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

Percentage Accurate: 99.3% → 99.3%

Time: 25.6s

Alternatives: 28

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (fma
   (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
   3.0
   (* (- 1.5 (* (sqrt 5.0) 0.5)) (* (cos y) 3.0)))))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((1.5 - (sqrt(5.0) * 0.5)) * (cos(y) * 3.0)));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(1.5 - Float64(sqrt(5.0) * 0.5)) * Float64(cos(y) * 3.0))))
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 3.0), $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. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

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

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

   5. div-sub N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. metadata-eval N/A

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

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

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

   16. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(3, \sqrt{5} \cdot 0.5, -1.5\right), 3\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (cos x) (cos y))
   (*
    (sqrt 2.0)
    (* (fma -0.0625 (sin y) (sin x)) (fma -0.0625 (sin x) (sin y))))
   2.0)
  (fma
   (fma (sqrt 5.0) -1.5 4.5)
   (cos y)
   (fma (cos x) (fma 3.0 (* (sqrt 5.0) 0.5) -1.5) 3.0))))

C

double code(double x, double y) {
	return fma((cos(x) - cos(y)), (sqrt(2.0) * (fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y)))), 2.0) / fma(fma(sqrt(5.0), -1.5, 4.5), cos(y), fma(cos(x), fma(3.0, (sqrt(5.0) * 0.5), -1.5), 3.0));
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(x) - cos(y)), Float64(sqrt(2.0) * Float64(fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y)))), 2.0) / fma(fma(sqrt(5.0), -1.5, 4.5), cos(y), fma(cos(x), fma(3.0, Float64(sqrt(5.0) * 0.5), -1.5), 3.0)))
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[(-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[(N[(N[Sqrt[5.0], $MachinePrecision] * -1.5 + 4.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] + -1.5), $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. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

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

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

   5. div-sub N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. metadata-eval N/A

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

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

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

   16. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.3%

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in x around inf

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), 3\right)\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. associate-+l+ N/A

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

    3. *-commutative N/A

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

    4. metadata-eval N/A

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

    5. distribute-lft-in N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

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

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

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

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

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

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

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

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

    12. *-commutative N/A

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

    13. distribute-lft-in N/A

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

    14. metadata-eval N/A

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

11. Applied egg-rr 99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(3, \sqrt{5} \cdot 0.5, -1.5\right), 3\right)\right)}} \]
12. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), 3\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- (cos x) (cos y))
   (*
    (sqrt 2.0)
    (* (fma -0.0625 (sin y) (sin x)) (fma -0.0625 (sin x) (sin y))))
   2.0)
  (fma
   (fma (sqrt 5.0) 1.5 -1.5)
   (cos x)
   (fma (cos y) (fma (sqrt 5.0) -1.5 4.5) 3.0))))

C

double code(double x, double y) {
	return fma((cos(x) - cos(y)), (sqrt(2.0) * (fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y)))), 2.0) / fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), fma(cos(y), fma(sqrt(5.0), -1.5, 4.5), 3.0));
}

Julia

function code(x, y)
	return Float64(fma(Float64(cos(x) - cos(y)), Float64(sqrt(2.0) * Float64(fma(-0.0625, sin(y), sin(x)) * fma(-0.0625, sin(x), sin(y)))), 2.0) / fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), fma(cos(y), fma(sqrt(5.0), -1.5, 4.5), 3.0)))
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[(-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[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + -1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -1.5 + 4.5), $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. accelerator-lowering-fma.f64 N/A

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

   3. +-commutative N/A

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

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

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

   5. div-sub N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

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

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

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

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

   14. metadata-eval N/A

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

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

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

   16. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.3%

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in x around inf

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), 3\right)\right)}} \]

10. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. associate-+l+ N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

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

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-in N/A

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

    9. *-commutative N/A

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

    10. associate-*l* N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

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

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

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

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

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

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

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

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

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

12. Simplified 99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), 3\right)\right)}} \]
13. Add Preprocessing

Alternative 4: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot 3\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{\mathsf{fma}\left(t\_1, \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), t\_0, 3\right)\right)}\\ \mathbf{if}\;x \leq -0.025:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(t\_2 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) 3.0))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (- (sin y) (/ (sin x) 16.0)))
        (t_3
         (/
          (+ 2.0 (* (- (cos x) (cos y)) (* t_2 (* (sqrt 2.0) (sin x)))))
          (fma t_1 (* (cos x) 3.0) (fma (fma (sqrt 5.0) -0.5 1.5) t_0 3.0)))))
   (if (<= x -0.025)
     t_3
     (if (<= x 5.2e-23)
       (/
        (+
         2.0
         (*
          (* t_2 (* (sqrt 2.0) (fma -0.0625 (sin y) x)))
          (fma x (* x -0.5) (- 1.0 (cos y)))))
        (fma (fma t_1 (cos x) 1.0) 3.0 (* (- 1.5 (* (sqrt 5.0) 0.5)) t_0)))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(y) * 3.0;
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = sin(y) - (sin(x) / 16.0);
	double t_3 = (2.0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0) * sin(x))))) / fma(t_1, (cos(x) * 3.0), fma(fma(sqrt(5.0), -0.5, 1.5), t_0, 3.0));
	double tmp;
	if (x <= -0.025) {
		tmp = t_3;
	} else if (x <= 5.2e-23) {
		tmp = (2.0 + ((t_2 * (sqrt(2.0) * fma(-0.0625, sin(y), x))) * fma(x, (x * -0.5), (1.0 - cos(y))))) / fma(fma(t_1, cos(x), 1.0), 3.0, ((1.5 - (sqrt(5.0) * 0.5)) * t_0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * 3.0)
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_3 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_2 * Float64(sqrt(2.0) * sin(x))))) / fma(t_1, Float64(cos(x) * 3.0), fma(fma(sqrt(5.0), -0.5, 1.5), t_0, 3.0)))
	tmp = 0.0
	if (x <= -0.025)
		tmp = t_3;
	elseif (x <= 5.2e-23)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))) * fma(x, Float64(x * -0.5), Float64(1.0 - cos(y))))) / fma(fma(t_1, cos(x), 1.0), 3.0, Float64(Float64(1.5 - Float64(sqrt(5.0) * 0.5)) * t_0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.025], t$95$3, If[LessEqual[x, 5.2e-23], N[(N[(2.0 + N[(N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.025000000000000001 or 5.2e-23 < x

   1. Initial program 98.8%

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

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

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

      3. +-commutative N/A

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

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

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.0%

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 64.1

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

   9. Simplified 64.1%

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

   if -0.025000000000000001 < x < 5.2e-23

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. sin-lowering-sin.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   7. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   10. Simplified 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.025:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1.5 - \sqrt{5} \cdot 0.5\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x - \cos y\\ t_3 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(t\_2 \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, t\_3, 1\right)\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, \cos x, 1\right), 3, t\_0 \cdot \left(\cos y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\mathsf{fma}\left(t\_0, \cos y, 1\right) + \cos x \cdot t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.5 (* (sqrt 5.0) 0.5)))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -0.06)
     (/
      (fma (sin x) (* (sqrt 2.0) (* t_2 (fma (sin x) -0.0625 (sin y)))) 2.0)
      (* 3.0 (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (fma (cos x) t_3 1.0))))
     (if (<= x 5.2e-23)
       (/
        (+
         2.0
         (*
          (* t_1 (* (sqrt 2.0) (fma -0.0625 (sin y) x)))
          (fma x (* x -0.5) (- 1.0 (cos y)))))
        (fma (fma t_3 (cos x) 1.0) 3.0 (* t_0 (* (cos y) 3.0))))
       (/
        (+ 2.0 (* t_2 (* t_1 (* (sqrt 2.0) (sin x)))))
        (* 3.0 (+ (fma t_0 (cos y) 1.0) (* (cos x) t_3))))))))

C

double code(double x, double y) {
	double t_0 = 1.5 - (sqrt(5.0) * 0.5);
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -0.06) {
		tmp = fma(sin(x), (sqrt(2.0) * (t_2 * fma(sin(x), -0.0625, sin(y)))), 2.0) / (3.0 * fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), t_3, 1.0)));
	} else if (x <= 5.2e-23) {
		tmp = (2.0 + ((t_1 * (sqrt(2.0) * fma(-0.0625, sin(y), x))) * fma(x, (x * -0.5), (1.0 - cos(y))))) / fma(fma(t_3, cos(x), 1.0), 3.0, (t_0 * (cos(y) * 3.0)));
	} else {
		tmp = (2.0 + (t_2 * (t_1 * (sqrt(2.0) * sin(x))))) / (3.0 * (fma(t_0, cos(y), 1.0) + (cos(x) * t_3)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.5 - Float64(sqrt(5.0) * 0.5))
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -0.06)
		tmp = Float64(fma(sin(x), Float64(sqrt(2.0) * Float64(t_2 * fma(sin(x), -0.0625, sin(y)))), 2.0) / Float64(3.0 * fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), t_3, 1.0))));
	elseif (x <= 5.2e-23)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))) * fma(x, Float64(x * -0.5), Float64(1.0 - cos(y))))) / fma(fma(t_3, cos(x), 1.0), 3.0, Float64(t_0 * Float64(cos(y) * 3.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_1 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(fma(t_0, cos(y), 1.0) + Float64(cos(x) * t_3))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.06], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $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] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-23], N[(N[(2.0 + N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.059999999999999998

   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

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

      2. associate-+r+ N/A

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

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

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

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

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

      5. div-sub N/A

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

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

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 98.7%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 60.7

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

   9. Simplified 60.7%

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

   if -0.059999999999999998 < x < 5.2e-23

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. sin-lowering-sin.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   7. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   10. Simplified 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   if 5.2e-23 < x

   1. Initial program 98.7%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

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

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

      5. div-sub N/A

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

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

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 67.0

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

   7. Simplified 67.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y, 1\right) + \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, t\_0, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.0205:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (/
          (fma
           (sin x)
           (* (sqrt 2.0) (* (- (cos x) (cos y)) (fma (sin x) -0.0625 (sin y))))
           2.0)
          (*
           3.0
           (fma (fma (sqrt 5.0) -0.5 1.5) (cos y) (fma (cos x) t_0 1.0))))))
   (if (<= x -0.0205)
     t_1
     (if (<= x 5.2e-23)
       (/
        (+
         2.0
         (*
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (fma -0.0625 (sin y) x)))
          (fma x (* x -0.5) (- 1.0 (cos y)))))
        (fma
         (fma t_0 (cos x) 1.0)
         3.0
         (* (- 1.5 (* (sqrt 5.0) 0.5)) (* (cos y) 3.0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = fma(sin(x), (sqrt(2.0) * ((cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y)))), 2.0) / (3.0 * fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), t_0, 1.0)));
	double tmp;
	if (x <= -0.0205) {
		tmp = t_1;
	} else if (x <= 5.2e-23) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(-0.0625, sin(y), x))) * fma(x, (x * -0.5), (1.0 - cos(y))))) / fma(fma(t_0, cos(x), 1.0), 3.0, ((1.5 - (sqrt(5.0) * 0.5)) * (cos(y) * 3.0)));
	} 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(x), Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y)))), 2.0) / Float64(3.0 * fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), t_0, 1.0))))
	tmp = 0.0
	if (x <= -0.0205)
		tmp = t_1;
	elseif (x <= 5.2e-23)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))) * fma(x, Float64(x * -0.5), Float64(1.0 - cos(y))))) / fma(fma(t_0, cos(x), 1.0), 3.0, Float64(Float64(1.5 - Float64(sqrt(5.0) * 0.5)) * Float64(cos(y) * 3.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(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 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0205], t$95$1, If[LessEqual[x, 5.2e-23], N[(N[(2.0 + N[(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] * N[(x * N[(x * -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0205000000000000009 or 5.2e-23 < x

   1. Initial program 98.8%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

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

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

      5. div-sub N/A

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

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

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in y around 0

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

      1. sin-lowering-sin.f64 64.0

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

   9. Simplified 64.0%

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

   if -0.0205000000000000009 < x < 5.2e-23

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. sin-lowering-sin.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   7. Simplified 99.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 99.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]

   10. Simplified 99.6%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0205:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := {\sin y}^{2}\\ t_2 := \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right)\\ \mathbf{if}\;y \leq -0.012:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right), 2\right)}{\mathsf{fma}\left(t\_0, \cos x \cdot 3, \mathsf{fma}\left(t\_2, \cos y \cdot 3, 3\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right)}{3 \cdot \mathsf{fma}\left(t\_2, \cos y, \mathsf{fma}\left(\cos x, t\_0, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \end{array} \]

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 (fma (sqrt 5.0) -0.5 1.5)))
   (if (<= y -0.012)
     (/
      (fma t_1 (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625)) 2.0)
      (fma t_0 (* (cos x) 3.0) (fma t_2 (* (cos y) 3.0) 3.0)))
     (if (<= y 1.5e-6)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (sqrt 2.0) (+ (cos x) -1.0)) (fma -0.0625 (sin x) y))
         2.0)
        (* 3.0 (fma t_2 (cos y) (fma (cos x) t_0 1.0))))
       (/
        (fma t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma
         1.5
         (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) (- 3.0 (sqrt 5.0))))
         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 = fma(sqrt(5.0), -0.5, 1.5);
	double tmp;
	if (y <= -0.012) {
		tmp = fma(t_1, ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)), 2.0) / fma(t_0, (cos(x) * 3.0), fma(t_2, (cos(y) * 3.0), 3.0));
	} else if (y <= 1.5e-6) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), ((sqrt(2.0) * (cos(x) + -1.0)) * fma(-0.0625, sin(x), y)), 2.0) / (3.0 * fma(t_2, cos(y), fma(cos(x), t_0, 1.0)));
	} else {
		tmp = fma(t_1, (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * (3.0 - sqrt(5.0)))), 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 = fma(sqrt(5.0), -0.5, 1.5)
	tmp = 0.0
	if (y <= -0.012)
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)), 2.0) / fma(t_0, Float64(cos(x) * 3.0), fma(t_2, Float64(cos(y) * 3.0), 3.0)));
	elseif (y <= 1.5e-6)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(sqrt(2.0) * Float64(cos(x) + -1.0)) * fma(-0.0625, sin(x), y)), 2.0) / Float64(3.0 * fma(t_2, cos(y), fma(cos(x), t_0, 1.0))));
	else
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * Float64(3.0 - sqrt(5.0)))), 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[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision]}, If[LessEqual[y, -0.012], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision] + N[(t$95$2 * N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-6], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 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] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.012

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.1%

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 60.6

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

   9. Simplified 60.6%

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

   if -0.012 < y < 1.5e-6

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

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

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

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

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

      5. div-sub N/A

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

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

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

      7. metadata-eval N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 98.9

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

   9. Simplified 98.9%

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

   if 1.5e-6 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   5. Simplified 63.2%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

   8. Simplified 63.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.012:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(y, -0.0625, \sin x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, 3\right) - \sqrt{5}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (pow (sin y) 2.0)))
   (if (<= y -0.0118)
     (/
      (fma t_1 (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625)) 2.0)
      (fma
       (fma (sqrt 5.0) 0.5 -0.5)
       (* (cos x) 3.0)
       (fma (fma (sqrt 5.0) -0.5 1.5) (* (cos y) 3.0) 3.0)))
     (if (<= y 1.5e-6)
       (/
        (+
         2.0
         (*
          (- (cos x) (cos y))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (fma y -0.0625 (sin x))))))
        (fma 1.5 (- (fma t_0 (cos x) 3.0) (sqrt 5.0)) 3.0))
       (/
        (fma t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos x) t_0 (* (cos y) (- 3.0 (sqrt 5.0)))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.0118) {
		tmp = fma(t_1, ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)), 2.0) / fma(fma(sqrt(5.0), 0.5, -0.5), (cos(x) * 3.0), fma(fma(sqrt(5.0), -0.5, 1.5), (cos(y) * 3.0), 3.0));
	} else if (y <= 1.5e-6) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(y, -0.0625, sin(x)))))) / fma(1.5, (fma(t_0, cos(x), 3.0) - sqrt(5.0)), 3.0);
	} else {
		tmp = fma(t_1, (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(x), t_0, (cos(y) * (3.0 - sqrt(5.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.0118)
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)), 2.0) / fma(fma(sqrt(5.0), 0.5, -0.5), Float64(cos(x) * 3.0), fma(fma(sqrt(5.0), -0.5, 1.5), Float64(cos(y) * 3.0), 3.0)));
	elseif (y <= 1.5e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(y, -0.0625, sin(x)))))) / fma(1.5, Float64(fma(t_0, cos(x), 3.0) - sqrt(5.0)), 3.0));
	else
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(cos(y) * Float64(3.0 - sqrt(5.0)))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.0118], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-6], 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[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0117999999999999997

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.1%

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 60.6

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

   9. Simplified 60.6%

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

   if -0.0117999999999999997 < y < 1.5e-6

   1. Initial program 99.4%

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

   3. 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 98.8%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. +-commutative N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. sin-lowering-sin.f64 98.8

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

   8. Simplified 98.8%

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

   if 1.5e-6 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   5. Simplified 63.2%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

   8. Simplified 63.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(y, -0.0625, \sin x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}\\ \mathbf{elif}\;y \leq 0.0006:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, \cos x \cdot t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \cos y \cdot t\_2\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.0118)
     (/
      (fma t_1 (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625)) 2.0)
      (fma
       (fma (sqrt 5.0) 0.5 -0.5)
       (* (cos x) 3.0)
       (fma (fma (sqrt 5.0) -0.5 1.5) (* (cos y) 3.0) 3.0)))
     (if (<= y 0.0006)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos y) t_2 (* (cos x) t_0)) 1.0))
       (/
        (fma t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos x) t_0 (* (cos y) t_2)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.0118) {
		tmp = fma(t_1, ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)), 2.0) / fma(fma(sqrt(5.0), 0.5, -0.5), (cos(x) * 3.0), fma(fma(sqrt(5.0), -0.5, 1.5), (cos(y) * 3.0), 3.0));
	} else if (y <= 0.0006) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(y), t_2, (cos(x) * t_0)), 1.0);
	} else {
		tmp = fma(t_1, (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(x), t_0, (cos(y) * t_2)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.0118)
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)), 2.0) / fma(fma(sqrt(5.0), 0.5, -0.5), Float64(cos(x) * 3.0), fma(fma(sqrt(5.0), -0.5, 1.5), Float64(cos(y) * 3.0), 3.0)));
	elseif (y <= 0.0006)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(y), t_2, Float64(cos(x) * t_0)), 1.0));
	else
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(cos(y) * t_2)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0118], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0006], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0117999999999999997

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

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

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

      16. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.1%

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

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

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

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

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

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

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

      16. sqrt-lowering-sqrt.f64 60.6

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

   9. Simplified 60.6%

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

   if -0.0117999999999999997 < y < 5.99999999999999947e-4

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*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 98.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 1\right)}} \]

   if 5.99999999999999947e-4 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   5. Simplified 62.7%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

   8. Simplified 62.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}\\ \mathbf{elif}\;y \leq 0.0006:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), t\_1, 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_2}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_2\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= y -0.0118)
     (/
      (fma (- 0.5 (* 0.5 (cos (+ y y)))) t_1 2.0)
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_2 2.0))) (* (cos y) (/ t_0 2.0)))))
     (if (<= y 0.0008)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos y) t_0 (* (cos x) t_2)) 1.0))
       (/
        (fma (pow (sin y) 2.0) t_1 2.0)
        (fma 1.5 (fma (cos x) t_2 (* (cos y) t_0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(2.0) * fma(cos(y), 0.0625, -0.0625);
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -0.0118) {
		tmp = fma((0.5 - (0.5 * cos((y + y)))), t_1, 2.0) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (y <= 0.0008) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(y), t_0, (cos(x) * t_2)), 1.0);
	} else {
		tmp = fma(pow(sin(y), 2.0), t_1, 2.0) / fma(1.5, fma(cos(x), t_2, (cos(y) * t_0)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -0.0118)
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), t_1, 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (y <= 0.0008)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(y), t_0, Float64(cos(x) * t_2)), 1.0));
	else
		tmp = Float64(fma((sin(y) ^ 2.0), t_1, 2.0) / fma(1.5, fma(cos(x), t_2, Float64(cos(y) * t_0)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -0.0118], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0008], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0117999999999999997

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   5. Simplified 60.5%

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

      1. unpow2 N/A

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

      2. sqr-sin-a N/A

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

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

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

      4. cos-2 N/A

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

      5. cos-sum N/A

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

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

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

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

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

      8. +-lowering-+.f64 60.5

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

   7. Applied egg-rr 60.5%

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

   if -0.0117999999999999997 < y < 8.00000000000000038e-4

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*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 98.4%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 1\right)}} \]

   if 8.00000000000000038e-4 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

   5. Simplified 62.7%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

   8. Simplified 62.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right)\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), t\_1, 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_2}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= y -0.0118)
     (/
      (fma (- 0.5 (* 0.5 (cos (+ y y)))) t_1 2.0)
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_2 2.0))) (* (cos y) (/ t_0 2.0)))))
     (if (<= y 1.5e-6)
       (*
        (fma
         (+ 0.5 (* -0.5 (cos (+ x x))))
         (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
         2.0)
        (/
         0.3333333333333333
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))))
       (/
        (fma (pow (sin y) 2.0) t_1 2.0)
        (fma 1.5 (fma (cos x) t_2 (* (cos y) t_0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(2.0) * fma(cos(y), 0.0625, -0.0625);
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -0.0118) {
		tmp = fma((0.5 - (0.5 * cos((y + y)))), t_1, 2.0) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (y <= 1.5e-6) {
		tmp = fma((0.5 + (-0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)));
	} else {
		tmp = fma(pow(sin(y), 2.0), t_1, 2.0) / fma(1.5, fma(cos(x), t_2, (cos(y) * t_0)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -0.0118)
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), t_1, 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (y <= 1.5e-6)
		tmp = Float64(fma(Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0))));
	else
		tmp = Float64(fma((sin(y) ^ 2.0), t_1, 2.0) / fma(1.5, fma(cos(x), t_2, Float64(cos(y) * t_0)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -0.0118], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-6], N[(N[(N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $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[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0117999999999999997

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 60.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin y \cdot \sin y}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. cos-2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \cos y - \sin y \cdot \sin y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. cos-sum N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-lowering-+.f64 60.5

         \[\leadsto \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 60.5%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -0.0117999999999999997 < y < 1.5e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. 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 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]

   if 1.5e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 63.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Simplified 63.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (- 0.5 (* 0.5 (cos (+ y y))))
           (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625))
           2.0)
          (*
           3.0
           (+
            (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
            (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))
   (if (<= y -0.0118)
     t_0
     (if (<= y 1.5e-6)
       (*
        (fma
         (+ 0.5 (* -0.5 (cos (+ x x))))
         (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
         2.0)
        (/
         0.3333333333333333
         (fma
          (fma (sqrt 5.0) -0.5 1.5)
          (cos y)
          (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((0.5 - (0.5 * cos((y + y)))), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	double tmp;
	if (y <= -0.0118) {
		tmp = t_0;
	} else if (y <= 1.5e-6) {
		tmp = fma((0.5 + (-0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 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)))))
	tmp = 0.0
	if (y <= -0.0118)
		tmp = t_0;
	elseif (y <= 1.5e-6)
		tmp = Float64(fma(Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $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[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0118], t$95$0, If[LessEqual[y, 1.5e-6], N[(N[(N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $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[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0117999999999999997 or 1.5e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 61.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin y \cdot \sin y}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. cos-2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \cos y - \sin y \cdot \sin y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. cos-sum N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-lowering-+.f64 61.9

         \[\leadsto \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \color{blue}{\left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(y + y\right)}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -0.0117999999999999997 < y < 1.5e-6

   1. Initial program 99.4%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. 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 98.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0118:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (+ 0.5 (* -0.5 (cos (+ x x))))
           (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
           2.0)
          (/
           0.3333333333333333
           (fma
            (fma (sqrt 5.0) -0.5 1.5)
            (cos y)
            (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))))))
   (if (<= x -4.2e-6)
     t_0
     (if (<= x 5.2e-23)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (+ (sqrt 5.0) -1.0)) 3.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((0.5 + (-0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * (0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)));
	double tmp;
	if (x <= -4.2e-6) {
		tmp = t_0;
	} else if (x <= 5.2e-23) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) + -1.0)), 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * Float64(0.3333333333333333 / fma(fma(sqrt(5.0), -0.5, 1.5), cos(y), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0))))
	tmp = 0.0
	if (x <= -4.2e-6)
		tmp = t_0;
	elseif (x <= 5.2e-23)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) + -1.0)), 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $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[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e-6], t$95$0, If[LessEqual[x, 5.2e-23], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999996e-6 or 5.2e-23 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      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 60.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied egg-rr 60.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]

   if -4.1999999999999996e-6 < x < 5.2e-23

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 3\right)} \]

      15. sqrt-lowering-sqrt.f64 99.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 3\right)} \]

   8. Simplified 99.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_2, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 1\right), 3, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right)\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot t\_2, 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(t\_0, \cos x, 2.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (pow (sin x) 2.0))
        (t_2 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))))
   (if (<= x -5.2e-6)
     (/
      (fma t_1 t_2 2.0)
      (fma (fma t_0 (cos x) 1.0) 3.0 (fma (sqrt 5.0) -1.5 4.5)))
     (if (<= x 5.2e-23)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (+ (sqrt 5.0) -1.0)) 3.0))
       (/
        (fma (* t_1 t_2) 0.3333333333333333 0.6666666666666666)
        (fma -0.5 (sqrt 5.0) (fma t_0 (cos x) 2.5)))))))

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 <= -5.2e-6) {
		tmp = fma(t_1, t_2, 2.0) / fma(fma(t_0, cos(x), 1.0), 3.0, fma(sqrt(5.0), -1.5, 4.5));
	} else if (x <= 5.2e-23) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) + -1.0)), 3.0);
	} else {
		tmp = fma((t_1 * t_2), 0.3333333333333333, 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(t_0, cos(x), 2.5));
	}
	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 <= -5.2e-6)
		tmp = Float64(fma(t_1, t_2, 2.0) / fma(fma(t_0, cos(x), 1.0), 3.0, fma(sqrt(5.0), -1.5, 4.5)));
	elseif (x <= 5.2e-23)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) + -1.0)), 3.0));
	else
		tmp = Float64(fma(Float64(t_1 * t_2), 0.3333333333333333, 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(t_0, cos(x), 2.5)));
	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[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, -5.2e-6], N[(N[(t$95$1 * t$95$2 + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[Sqrt[5.0], $MachinePrecision] * -1.5 + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-23], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000019e-6

   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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1}, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)}, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)}\right)} \]

   4. Applied egg-rr 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + 1\right) \cdot 3 + \color{blue}{\left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3}} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + 1\right) + \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y\right)}} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + \left(1 + \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + \color{blue}{\left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}, \cos x \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}, \cos x \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{-1}{2}\right), \cos x \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x \cdot 3}, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x} \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

   6. Applied egg-rr 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)}} \]

   9. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right)\right)}} \]

   if -5.20000000000000019e-6 < x < 5.2e-23

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 3\right)} \]

      15. sqrt-lowering-sqrt.f64 99.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 3\right)} \]

   8. Simplified 99.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]

   if 5.2e-23 < x

   1. Initial program 98.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5}} \cdot \frac{1}{2}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y, 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)}} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{5}{2} + \left(\frac{-1}{2} \cdot \sqrt{5} + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 2.5\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, t\_2\right), 1\right)}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_2, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 2.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -7.8e-6)
     (*
      0.3333333333333333
      (/
       (fma t_1 (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)) 2.0)
       (fma 0.5 (fma (cos x) t_0 t_2) 1.0)))
     (if (<= x 5.2e-23)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) t_2 t_0) 3.0))
       (/
        (fma
         (* t_1 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma -0.5 (sqrt 5.0) (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 2.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = pow(sin(x), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -7.8e-6) {
		tmp = 0.3333333333333333 * (fma(t_1, (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(0.5, fma(cos(x), t_0, t_2), 1.0));
	} else if (x <= 5.2e-23) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_2, t_0), 3.0);
	} else {
		tmp = fma((t_1 * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 2.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -7.8e-6)
		tmp = Float64(0.3333333333333333 * Float64(fma(t_1, Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(0.5, fma(cos(x), t_0, t_2), 1.0)));
	elseif (x <= 5.2e-23)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_2, t_0), 3.0));
	else
		tmp = Float64(fma(Float64(t_1 * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(-0.5, sqrt(5.0), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 2.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e-6], N[(0.3333333333333333 * N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-23], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.7999999999999999e-6

   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 56.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Simplified 55.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]

   if -7.7999999999999999e-6 < x < 5.2e-23

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 3\right)} \]

      15. sqrt-lowering-sqrt.f64 99.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 3\right)} \]

   8. Simplified 99.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]

   if 5.2e-23 < x

   1. Initial program 98.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2}, \cos y, 1\right)} + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{3}{2} - \frac{\sqrt{5}}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \color{blue}{\sqrt{5}} \cdot \frac{1}{2}, \cos y, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \color{blue}{\cos y}, 1\right) + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\mathsf{fma}\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}, \cos y, 1\right) + \color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x}\right)} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(1.5 - \sqrt{5} \cdot 0.5, \cos y, 1\right) + \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right)}} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{5}{2} + \left(\frac{-1}{2} \cdot \sqrt{5} + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(-0.5, \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 2.5\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_1, t\_0\right), 1\right)}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (*
          0.3333333333333333
          (/
           (fma
            (pow (sin x) 2.0)
            (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625))
            2.0)
           (fma 0.5 (fma (cos x) t_1 t_0) 1.0)))))
   (if (<= x -3.5e-5)
     t_2
     (if (<= x 5.2e-23)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) t_0 t_1) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = 0.3333333333333333 * (fma(pow(sin(x), 2.0), (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(0.5, fma(cos(x), t_1, t_0), 1.0));
	double tmp;
	if (x <= -3.5e-5) {
		tmp = t_2;
	} else if (x <= 5.2e-23) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, t_1), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(0.3333333333333333 * Float64(fma((sin(x) ^ 2.0), Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(0.5, fma(cos(x), t_1, t_0), 1.0)))
	tmp = 0.0
	if (x <= -3.5e-5)
		tmp = t_2;
	elseif (x <= 5.2e-23)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, t_1), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e-5], t$95$2, If[LessEqual[x, 5.2e-23], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999997e-5 or 5.2e-23 < x

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      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 60.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Simplified 59.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]

   if -3.4999999999999997e-5 < x < 5.2e-23

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 3\right)} \]

      15. sqrt-lowering-sqrt.f64 99.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 3\right)} \]

   8. Simplified 99.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, 3\right) - \sqrt{5}, 1.5, 3\right)}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot t\_0, 2\right)}}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot t\_0, 2\right)}{\mathsf{fma}\left(\sqrt{5}, -1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 7.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (cos x) -0.0625 0.0625)) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (<= x -2.3e-5)
     (/
      1.0
      (/
       (fma (- (fma (cos x) t_1 3.0) (sqrt 5.0)) 1.5 3.0)
       (fma (- 0.5 (* 0.5 (cos (+ x x)))) (* (sqrt 2.0) t_0) 2.0)))
     (if (<= x 5.2e-23)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_1) 3.0))
       (/
        (fma (sqrt 2.0) (* (pow (sin x) 2.0) t_0) 2.0)
        (fma (sqrt 5.0) -1.5 (fma (fma (sqrt 5.0) 1.5 -1.5) (cos x) 7.5)))))))

C

double code(double x, double y) {
	double t_0 = fma(cos(x), -0.0625, 0.0625);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -2.3e-5) {
		tmp = 1.0 / (fma((fma(cos(x), t_1, 3.0) - sqrt(5.0)), 1.5, 3.0) / fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * t_0), 2.0));
	} else if (x <= 5.2e-23) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), t_1), 3.0);
	} else {
		tmp = fma(sqrt(2.0), (pow(sin(x), 2.0) * t_0), 2.0) / fma(sqrt(5.0), -1.5, fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 7.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(cos(x), -0.0625, 0.0625)
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -2.3e-5)
		tmp = Float64(1.0 / Float64(fma(Float64(fma(cos(x), t_1, 3.0) - sqrt(5.0)), 1.5, 3.0) / fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * t_0), 2.0)));
	elseif (x <= 5.2e-23)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_1), 3.0));
	else
		tmp = Float64(fma(sqrt(2.0), Float64((sin(x) ^ 2.0) * t_0), 2.0) / fma(sqrt(5.0), -1.5, fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 7.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -2.3e-5], N[(1.0 / N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $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$0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-23], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * -1.5 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + -1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 7.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3e-5

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 56.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      9. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right), 3 - \sqrt{5}\right), 3\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

      13. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      14. sqrt-lowering-sqrt.f64 55.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Simplified 55.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]

   10. Applied egg-rr 55.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 1.5, 3\right)}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}} \]

   if -2.3e-5 < x < 5.2e-23

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. --lowering--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + -1}\right), 3\right)} \]

      15. sqrt-lowering-sqrt.f64 99.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + -1\right), 3\right)} \]

   8. Simplified 99.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]

   if 5.2e-23 < x

   1. Initial program 98.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. 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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1}, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{\sqrt{5} - 1}{2}, \cos x, 1\right)}, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      5. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} - \frac{1}{2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\sqrt{5}}{2} - \color{blue}{\frac{1}{2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\sqrt{5}}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{1}{-2}}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{1}{-2}\right)}, \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{1}{-2}\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \color{blue}{\frac{-1}{2}}\right), \cos x, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x}, 1\right), 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)}\right)} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 - \sqrt{5} \cdot 0.5\right) \cdot \left(\cos y \cdot 3\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + 1\right) \cdot 3 + \color{blue}{\left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y\right) \cdot 3}} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + 1\right) + \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y\right)}} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + \left(1 + \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x + \color{blue}{\left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right)}\right)} \]

      5. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}, \cos x \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)}, \cos x \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{5}}, \frac{1}{2}, \frac{-1}{2}\right), \cos x \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x \cdot 3}, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \color{blue}{\cos x} \cdot 3, \left(\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \cos y + 1\right) \cdot 3\right)} \]

   6. Applied egg-rr 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, 3\right)\right)}} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{1}{16} \cdot \sin y\right) \cdot \left(\sin y - \frac{1}{16} \cdot \sin x\right)\right)\right)}{3 + \left(3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot \left(\cos y \cdot \left(\frac{3}{2} + \frac{-1}{2} \cdot \sqrt{5}\right)\right)\right)}} \]

   9. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x, \mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -1.5, 4.5\right), 3\right)\right)}} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{15}{2} + \left(\frac{-3}{2} \cdot \sqrt{5} + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]

   12. Simplified 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, {\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\sqrt{5}, -1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 7.5\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 59.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (fma (cos x) -0.0625 0.0625)
   (* (sqrt 2.0) (- 0.5 (* 0.5 (cos (+ x x)))))
   2.0)
  (fma 3.0 (* 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (- 3.0 (sqrt 5.0)))) 3.0)))

C

double code(double x, double y) {
	return fma(fma(cos(x), -0.0625, 0.0625), (sqrt(2.0) * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(3.0, (0.5 * fma(cos(x), (sqrt(5.0) + -1.0), (3.0 - sqrt(5.0)))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(fma(cos(x), -0.0625, 0.0625), Float64(sqrt(2.0) * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(3.0, Float64(0.5 * fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(3.0 - sqrt(5.0)))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right), 3 - \sqrt{5}\right), 3\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. --lowering--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   14. sqrt-lowering-sqrt.f64 59.2

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Simplified 59.2%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x \cdot \frac{-1}{16} + \frac{1}{16}\right)} + 2}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\cos x \cdot \frac{-1}{16} + \frac{1}{16}\right) \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)} + 2}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x \cdot \frac{-1}{16} + \frac{1}{16}, {\sin x}^{2} \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, {\sin x}^{2} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \frac{-1}{16}, \frac{1}{16}\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \sqrt{2} \cdot \color{blue}{\left(\sin x \cdot \sin x\right)}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   10. sqr-sin-a N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   11. --lowering--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)\right)}, 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   12. cos-2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos x \cdot \cos x - \sin x \cdot \sin x\right)}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   13. cos-sum N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(x + x\right)}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \sqrt{2} \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(x + x\right)}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   15. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), \sqrt{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(x + x\right)}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   16. +-lowering-+.f64 59.2

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(x + x\right)}\right), 2\right)}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

10. Applied egg-rr 59.2%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
11. Add Preprocessing

Alternative 19: 59.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 1.5, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (fma
   (- 0.5 (* 0.5 (cos (+ x x))))
   (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
   2.0)
  (/ 1.0 (fma (- (fma (cos x) (+ (sqrt 5.0) -1.0) 3.0) (sqrt 5.0)) 1.5 3.0))))

C

double code(double x, double y) {
	return fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * (1.0 / fma((fma(cos(x), (sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 1.5, 3.0));
}

Julia

function code(x, y)
	return Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * Float64(1.0 / fma(Float64(fma(cos(x), Float64(sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 1.5, 3.0)))
end

Wolfram

code[x_, y_] := N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 1.5 + 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. 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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right), 3 - \sqrt{5}\right), 3\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. --lowering--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   14. sqrt-lowering-sqrt.f64 59.2

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Simplified 59.2%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]

10. Applied egg-rr 59.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 1.5, 3\right)}} \]
11. Add Preprocessing

Alternative 20: 59.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 1.5, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- 0.5 (* 0.5 (cos (+ x x))))
   (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
   2.0)
  (fma (- (fma (cos x) (+ (sqrt 5.0) -1.0) 3.0) (sqrt 5.0)) 1.5 3.0)))

C

double code(double x, double y) {
	return fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / fma((fma(cos(x), (sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 1.5, 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / fma(Float64(fma(cos(x), Float64(sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 1.5, 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 1.5 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right), 3 - \sqrt{5}\right), 3\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. --lowering--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   14. sqrt-lowering-sqrt.f64 59.2

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Simplified 59.2%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]

10. Applied egg-rr 59.2%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 1.5, 3\right)}} \]
11. Add Preprocessing

Alternative 21: 44.9% 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 x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. cos-lowering-cos.f64 63.2

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 63.2%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

8. Simplified 45.4%

   \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Final simplification 45.4%

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
10. Add Preprocessing

Alternative 22: 44.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot 3, 3\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (fma (sqrt 5.0) -0.5 1.5)
   (* (cos y) 3.0)
   (fma (cos x) (* (fma (sqrt 5.0) 0.5 -0.5) 3.0) 3.0))))

C

double code(double x, double y) {
	return 2.0 / fma(fma(sqrt(5.0), -0.5, 1.5), (cos(y) * 3.0), fma(cos(x), (fma(sqrt(5.0), 0.5, -0.5) * 3.0), 3.0));
}

Julia

function code(x, y)
	return Float64(2.0 / fma(fma(sqrt(5.0), -0.5, 1.5), Float64(cos(y) * 3.0), fma(cos(x), Float64(fma(sqrt(5.0), 0.5, -0.5) * 3.0), 3.0)))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * 3.0), $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. 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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \left(\cos y \cdot 3\right)} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

   4. div-sub N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)} \cdot \left(\cos y \cdot 3\right) + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\left(\color{blue}{\frac{3}{2}} - \frac{\sqrt{5}}{2}\right) \cdot \left(\cos y \cdot 3\right) + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

   6. div-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right) \cdot \left(\cos y \cdot 3\right) + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\cos y \cdot 3\right) + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot 3\right) + \color{blue}{3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot 3\right) + 3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)}} \]

   10. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot 3\right) + \color{blue}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + 1 \cdot 3\right)}} \]

7. Applied egg-rr 61.5%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot 3, 3\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{-1}{2}, \frac{3}{2}\right), \cos y \cdot 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot 3, 3\right)\right)} \]
9. Step-by-step derivation

10. Simplified 45.4%

    \[\leadsto \frac{\color{blue}{2}}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \cos y \cdot 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot 3, 3\right)\right)} \]
11. Add Preprocessing

Alternative 23: 42.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma 3.0 (* 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (- 3.0 (sqrt 5.0)))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(3.0, (0.5 * fma(cos(x), (sqrt(5.0) + -1.0), (3.0 - sqrt(5.0)))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(3.0, Float64(0.5 * fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(3.0 - sqrt(5.0)))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right), 3 - \sqrt{5}\right), 3\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. --lowering--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   14. sqrt-lowering-sqrt.f64 59.2

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Simplified 59.2%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{\mathsf{fma}\left(3, \frac{1}{2} \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Simplified 43.0%

    \[\leadsto \frac{\color{blue}{2}}{\mathsf{fma}\left(3, 0.5 \cdot \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
12. Add Preprocessing

Alternative 24: 42.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{-0.5 + \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (+ -0.5 (+ 1.0 (* 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)))))))

C

double code(double x, double y) {
	return 0.6666666666666666 / (-0.5 + (1.0 + (0.5 * fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)))));
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / Float64(-0.5 + Float64(1.0 + Float64(0.5 * fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0))))))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(-0.5 + N[(1.0 + N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $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

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right)} \]

   6. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   10. --lowering--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   13. metadata-eval 42.5

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + \color{blue}{-1}, 1\right)} \]

8. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + -1, 1\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{1 + \frac{1}{2} \cdot \left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2} + -1 \cdot \frac{1}{2}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{3}}{1 + \left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right)} \]

   4. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(1 + \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2}\right) + \frac{-1}{2}}} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(1 + \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2}\right) + \frac{-1}{2}}} \]

10. Applied egg-rr 42.5%

    \[\leadsto \frac{0.6666666666666666}{\color{blue}{\left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)\right) + -0.5}} \]

11. Final simplification 42.5%

    \[\leadsto \frac{0.6666666666666666}{-0.5 + \left(1 + 0.5 \cdot \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)\right)} \]
12. Add Preprocessing

Alternative 25: 42.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ 0.6666666666666666 \cdot \frac{1}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.6666666666666666
  (/ 1.0 (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) 0.5))))

C

double code(double x, double y) {
	return 0.6666666666666666 * (1.0 / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), 0.5));
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 * Float64(1.0 / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), 0.5)))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 * N[(1.0 / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right)} \]

   6. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   10. --lowering--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   13. metadata-eval 42.5

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + \color{blue}{-1}, 1\right)} \]

8. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + -1, 1\right)}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{2} \cdot \left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1\right) + 1}{\frac{2}{3}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{2} \cdot \left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1\right) + 1} \cdot \frac{2}{3}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{2} \cdot \left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1\right) + 1} \cdot \frac{2}{3}} \]

10. Applied egg-rr 42.5%

    \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \cdot 0.6666666666666666} \]

11. Final simplification 42.5%

    \[\leadsto 0.6666666666666666 \cdot \frac{1}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \]
12. Add Preprocessing

Alternative 26: 42.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} + -1\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) (+ (sqrt 5.0) -1.0)) 1.0)))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), (sqrt(5.0) + -1.0)), 1.0);
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(sqrt(5.0) + -1.0)), 1.0))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right)} \]

   6. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   10. --lowering--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   13. metadata-eval 42.5

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + \color{blue}{-1}, 1\right)} \]

8. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + -1, 1\right)}} \]
9. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right)}, 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} + -1\right), 1\right)} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} + -1\right)}, 1\right)} \]

   4. --lowering--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{\color{blue}{{5}^{1}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{{5}^{\color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   7. pow-prod-up N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{\color{blue}{{5}^{\frac{1}{2}} \cdot {5}^{\frac{1}{2}}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   8. pow1/2 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{\color{blue}{\sqrt{5}} \cdot {5}^{\frac{1}{2}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   9. pow1/2 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{\sqrt{5} \cdot \color{blue}{\sqrt{5}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{\sqrt{5} \cdot \sqrt{5}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   11. pow1/2 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{\color{blue}{{5}^{\frac{1}{2}}} \cdot \sqrt{5}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   12. pow1/2 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{{5}^{\frac{1}{2}} \cdot \color{blue}{{5}^{\frac{1}{2}}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   13. pow-prod-up N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{\color{blue}{{5}^{\left(\frac{1}{2} + \frac{1}{2}\right)}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{{5}^{\color{blue}{1}}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{\color{blue}{5}}, \cos y, \sqrt{5} + -1\right), 1\right)} \]

   16. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} + -1\right), 1\right)} \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} + -1}\right), 1\right)} \]

10. Applied egg-rr 42.5%

    \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} + -1\right)}, 1\right)} \]
11. Add Preprocessing

Alternative 27: 42.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5, 0.5\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) 0.5 0.5)))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), 0.5, 0.5);
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), 0.5, 0.5))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right)} \]

   6. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   10. --lowering--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   13. metadata-eval 42.5

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + \color{blue}{-1}, 1\right)} \]

8. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + -1, 1\right)}} \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2} + -1 \cdot \frac{1}{2}\right)} + 1} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) + 1} \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2} + \left(\frac{-1}{2} + 1\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{2}{3}}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}, \frac{1}{2}, \frac{1}{2}\right)}} \]

10. Applied egg-rr 42.5%

    \[\leadsto \frac{0.6666666666666666}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5, 0.5\right)}} \]
11. Add Preprocessing

Alternative 28: 40.1% 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 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 61.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}, 1\right)} \]

   6. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   10. --lowering--.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right) + \left(\mathsf{neg}\left(1\right)\right), 1\right)} \]

   13. metadata-eval 42.5

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + \color{blue}{-1}, 1\right)} \]

8. Simplified 42.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + -1, 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3}} \]
10. Step-by-step derivation

11. Simplified 40.5%

    \[\leadsto \color{blue}{0.3333333333333333} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))))