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

Percentage Accurate: 99.3% → 99.3%

Time: 24.2s

Alternatives: 46

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

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

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)))
    (* (/ 2.0 (+ (sqrt 5.0) 3.0)) (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))) + ((2.0 / (sqrt(5.0) + 3.0)) * 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))) + ((2.0d0 / (sqrt(5.0d0) + 3.0d0)) * 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))) + ((2.0 / (Math.sqrt(5.0) + 3.0)) * 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))) + ((2.0 / (math.sqrt(5.0) + 3.0)) * 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(2.0 / Float64(sqrt(5.0) + 3.0)) * 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))) + ((2.0 / (sqrt(5.0) + 3.0)) * 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[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, -3, 14\right), \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (fma -0.0625 (sin y) (sin x))
   (* (* (- (cos x) (cos y)) (sqrt 2.0)) (fma -0.0625 (sin x) (sin y)))
   2.0)
  (*
   3.0
   (fma
    (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
    (fma (sqrt 5.0) -3.0 14.0)
    (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))))

C

double code(double x, double y) {
	return fma(fma(-0.0625, sin(y), sin(x)), (((cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(x), sin(y))), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), fma(sqrt(5.0), -3.0, 14.0), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
}

Julia

function code(x, y)
	return Float64(fma(fma(-0.0625, sin(y), sin(x)), Float64(Float64(Float64(cos(x) - cos(y)) * sqrt(2.0)) * fma(-0.0625, sin(x), sin(y))), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), fma(sqrt(5.0), -3.0, 14.0), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))))
end

Wolfram

code[x_, y_] := N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -3.0 + 14.0), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.4

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

8. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \color{blue}{14 - 3 \cdot \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \color{blue}{14 + \left(\mathsf{neg}\left(3 \cdot \sqrt{5}\right)\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \color{blue}{\left(\mathsf{neg}\left(3 \cdot \sqrt{5}\right)\right) + 14}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \left(\mathsf{neg}\left(\color{blue}{3 \cdot \sqrt{5}}\right)\right) + 14, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \left(\mathsf{neg}\left(\color{blue}{\sqrt{5} \cdot 3}\right)\right) + 14, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \color{blue}{\sqrt{5} \cdot \left(\mathsf{neg}\left(3\right)\right)} + 14, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \color{blue}{\mathsf{fma}\left(\sqrt{5}, \mathsf{neg}\left(3\right), 14\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

   8. metadata-eval 99.4

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \mathsf{fma}\left(\sqrt{5}, \color{blue}{-3}, 14\right), \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, \color{blue}{\mathsf{fma}\left(\sqrt{5}, -3, 14\right)}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
11. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, 1\right)\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (*
     (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y)))
     (fma -0.0625 (sin y) (sin x)))
    (sqrt 2.0)
    2.0)
   (fma
    (/ (cos y) (+ (sqrt 5.0) 3.0))
    2.0
    (fma (* 0.5 (cos x)) (- (sqrt 5.0) 1.0) 1.0)))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma(((fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y))) * fma(-0.0625, sin(y), sin(x))), sqrt(2.0), 2.0) / fma((cos(y) / (sqrt(5.0) + 3.0)), 2.0, fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), 1.0))) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y))) * fma(-0.0625, sin(y), sin(x))), sqrt(2.0), 2.0) / fma(Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), 2.0, fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), 1.0))) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sqrt.f64 57.6

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

7. Applied rewrites 57.6%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 44.3%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 99.4%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma
    (*
     (* (fma -0.0625 (sin x) (sin y)) (- (cos x) (cos y)))
     (fma -0.0625 (sin y) (sin x)))
    (sqrt 2.0)
    2.0)
   (fma
    (fma (- 3.0 (sqrt 5.0)) (cos y) (* (- (sqrt 5.0) 1.0) (cos x)))
    0.5
    1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma(((fma(-0.0625, sin(x), sin(y)) * (cos(x) - cos(y))) * fma(-0.0625, sin(y), sin(x))), sqrt(2.0), 2.0) / fma(fma((3.0 - sqrt(5.0)), cos(y), ((sqrt(5.0) - 1.0) * cos(x))), 0.5, 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64(Float64(fma(-0.0625, sin(x), sin(y)) * Float64(cos(x) - cos(y))) * fma(-0.0625, sin(y), sin(x))), sqrt(2.0), 2.0) / fma(fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 0.5, 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. sub-neg N/A

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

   8. distribute-lft-in N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

5. Applied rewrites 45.5%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. associate-*r* N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 45.6%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.3%

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

Alternative 6: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)\\ t_1 := \sqrt{2} \cdot \left(\cos x - \cos y\right)\\ \mathbf{if}\;y \leq -0.186 \lor \neg \left(y \leq 0.51\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          0.5
          (fma (cos y) (- 3.0 (sqrt 5.0)) (* (cos x) (- (sqrt 5.0) 1.0)))
          1.0))
        (t_1 (* (sqrt 2.0) (- (cos x) (cos y)))))
   (if (or (<= y -0.186) (not (<= y 0.51)))
     (*
      (/ (fma t_1 (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0) t_0)
      0.3333333333333333)
     (*
      (/
       (fma
        t_1
        (*
         (- (sin y) (* 0.0625 (sin x)))
         (fma
          (fma
           (fma (* y y) -0.0005208333333333333 0.010416666666666666)
           (* y y)
           -0.0625)
          y
          (sin x)))
        2.0)
       t_0)
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), (cos(x) * (sqrt(5.0) - 1.0))), 1.0);
	double t_1 = sqrt(2.0) * (cos(x) - cos(y));
	double tmp;
	if ((y <= -0.186) || !(y <= 0.51)) {
		tmp = (fma(t_1, (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / t_0) * 0.3333333333333333;
	} else {
		tmp = (fma(t_1, ((sin(y) - (0.0625 * sin(x))) * fma(fma(fma((y * y), -0.0005208333333333333, 0.010416666666666666), (y * y), -0.0625), y, sin(x))), 2.0) / t_0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(cos(x) * Float64(sqrt(5.0) - 1.0))), 1.0)
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) - cos(y)))
	tmp = 0.0
	if ((y <= -0.186) || !(y <= 0.51))
		tmp = Float64(Float64(fma(t_1, Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / t_0) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(t_1, Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * fma(fma(fma(Float64(y * y), -0.0005208333333333333, 0.010416666666666666), Float64(y * y), -0.0625), y, sin(x))), 2.0) / t_0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.186], N[Not[LessEqual[y, 0.51]], $MachinePrecision]], N[(N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * -0.0005208333333333333 + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.186 or 0.51000000000000001 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.4%

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

   if -0.186 < y < 0.51000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.186 \lor \neg \left(y \leq 0.51\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, -0.0005208333333333333, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.186 \lor \neg \left(y \leq 0.51\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_0, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, \cos x \cdot t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos y, t\_1 \cdot \cos x\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -0.186) (not (<= y 0.51)))
     (*
      (/
       (fma (* (sqrt 2.0) t_0) (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0)
       (fma 0.5 (fma (cos y) t_2 (* (cos x) t_1)) 1.0))
      0.3333333333333333)
     (*
      (fma
       t_0
       (*
        (fma -0.0625 (sin x) (sin y))
        (*
         (fma
          (fma
           (fma -0.0005208333333333333 (* y y) 0.010416666666666666)
           (* y y)
           -0.0625)
          y
          (sin x))
         (sqrt 2.0)))
       2.0)
      (/
       0.3333333333333333
       (fma (fma t_2 (cos y) (* t_1 (cos x))) 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -0.186) || !(y <= 0.51)) {
		tmp = (fma((sqrt(2.0) * t_0), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(0.5, fma(cos(y), t_2, (cos(x) * t_1)), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(t_0, (fma(-0.0625, sin(x), sin(y)) * (fma(fma(fma(-0.0005208333333333333, (y * y), 0.010416666666666666), (y * y), -0.0625), y, sin(x)) * sqrt(2.0))), 2.0) * (0.3333333333333333 / fma(fma(t_2, cos(y), (t_1 * cos(x))), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -0.186) || !(y <= 0.51))
		tmp = Float64(Float64(fma(Float64(sqrt(2.0) * t_0), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(0.5, fma(cos(y), t_2, Float64(cos(x) * t_1)), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(t_0, Float64(fma(-0.0625, sin(x), sin(y)) * Float64(fma(fma(fma(-0.0005208333333333333, Float64(y * y), 0.010416666666666666), Float64(y * y), -0.0625), y, sin(x)) * sqrt(2.0))), 2.0) * Float64(0.3333333333333333 / fma(fma(t_2, cos(y), Float64(t_1 * cos(x))), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.186], N[Not[LessEqual[y, 0.51]], $MachinePrecision]], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(t$95$0 * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.0005208333333333333 * N[(y * y), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.186 or 0.51000000000000001 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.4%

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

   if -0.186 < y < 0.51000000000000001

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-sin.f64 99.5

           \[\leadsto \mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \color{blue}{\sin x}\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)} \]

   11. Applied rewrites 99.5%

       \[\leadsto \mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right)} \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.186 \lor \neg \left(y \leq 0.51\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)\\ t_1 := \sqrt{2} \cdot \left(\cos x - \cos y\right)\\ \mathbf{if}\;y \leq -0.155 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          0.5
          (fma (cos y) (- 3.0 (sqrt 5.0)) (* (cos x) (- (sqrt 5.0) 1.0)))
          1.0))
        (t_1 (* (sqrt 2.0) (- (cos x) (cos y)))))
   (if (or (<= y -0.155) (not (<= y 0.245)))
     (*
      (/ (fma t_1 (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0) t_0)
      0.3333333333333333)
     (*
      (/
       (fma
        t_1
        (*
         (- (sin y) (* 0.0625 (sin x)))
         (fma (fma (* y y) 0.010416666666666666 -0.0625) y (sin x)))
        2.0)
       t_0)
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), (cos(x) * (sqrt(5.0) - 1.0))), 1.0);
	double t_1 = sqrt(2.0) * (cos(x) - cos(y));
	double tmp;
	if ((y <= -0.155) || !(y <= 0.245)) {
		tmp = (fma(t_1, (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / t_0) * 0.3333333333333333;
	} else {
		tmp = (fma(t_1, ((sin(y) - (0.0625 * sin(x))) * fma(fma((y * y), 0.010416666666666666, -0.0625), y, sin(x))), 2.0) / t_0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(cos(x) * Float64(sqrt(5.0) - 1.0))), 1.0)
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) - cos(y)))
	tmp = 0.0
	if ((y <= -0.155) || !(y <= 0.245))
		tmp = Float64(Float64(fma(t_1, Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / t_0) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(t_1, Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * fma(fma(Float64(y * y), 0.010416666666666666, -0.0625), y, sin(x))), 2.0) / t_0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.155], N[Not[LessEqual[y, 0.245]], $MachinePrecision]], N[(N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.010416666666666666 + -0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.154999999999999999 or 0.245 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.4%

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

   if -0.154999999999999999 < y < 0.245

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.155 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.155 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_0, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, \cos x \cdot t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos y, t\_1 \cdot \cos x\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -0.155) (not (<= y 0.245)))
     (*
      (/
       (fma (* (sqrt 2.0) t_0) (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0)
       (fma 0.5 (fma (cos y) t_2 (* (cos x) t_1)) 1.0))
      0.3333333333333333)
     (*
      (fma
       t_0
       (*
        (fma -0.0625 (sin x) (sin y))
        (*
         (fma (fma (* y y) 0.010416666666666666 -0.0625) y (sin x))
         (sqrt 2.0)))
       2.0)
      (/
       0.3333333333333333
       (fma (fma t_2 (cos y) (* t_1 (cos x))) 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -0.155) || !(y <= 0.245)) {
		tmp = (fma((sqrt(2.0) * t_0), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(0.5, fma(cos(y), t_2, (cos(x) * t_1)), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(t_0, (fma(-0.0625, sin(x), sin(y)) * (fma(fma((y * y), 0.010416666666666666, -0.0625), y, sin(x)) * sqrt(2.0))), 2.0) * (0.3333333333333333 / fma(fma(t_2, cos(y), (t_1 * cos(x))), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -0.155) || !(y <= 0.245))
		tmp = Float64(Float64(fma(Float64(sqrt(2.0) * t_0), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(0.5, fma(cos(y), t_2, Float64(cos(x) * t_1)), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(t_0, Float64(fma(-0.0625, sin(x), sin(y)) * Float64(fma(fma(Float64(y * y), 0.010416666666666666, -0.0625), y, sin(x)) * sqrt(2.0))), 2.0) * Float64(0.3333333333333333 / fma(fma(t_2, cos(y), Float64(t_1 * cos(x))), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.155], N[Not[LessEqual[y, 0.245]], $MachinePrecision]], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(t$95$0 * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.010416666666666666 + -0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.154999999999999999 or 0.245 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.4%

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

   if -0.154999999999999999 < y < 0.245

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 99.4

           \[\leadsto \mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \color{blue}{\sin x}\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)} \]

   11. Applied rewrites 99.4%

       \[\leadsto \mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right)} \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.155 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)\\ t_1 := \sqrt{2} \cdot \left(\cos x - \cos y\right)\\ \mathbf{if}\;y \leq -0.065 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          0.5
          (fma (cos y) (- 3.0 (sqrt 5.0)) (* (cos x) (- (sqrt 5.0) 1.0)))
          1.0))
        (t_1 (* (sqrt 2.0) (- (cos x) (cos y)))))
   (if (or (<= y -0.065) (not (<= y 0.23)))
     (*
      (/ (fma t_1 (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0) t_0)
      0.3333333333333333)
     (*
      (/
       (fma t_1 (* (- (sin y) (* 0.0625 (sin x))) (fma -0.0625 y (sin x))) 2.0)
       t_0)
      0.3333333333333333))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), (cos(x) * (sqrt(5.0) - 1.0))), 1.0);
	double t_1 = sqrt(2.0) * (cos(x) - cos(y));
	double tmp;
	if ((y <= -0.065) || !(y <= 0.23)) {
		tmp = (fma(t_1, (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / t_0) * 0.3333333333333333;
	} else {
		tmp = (fma(t_1, ((sin(y) - (0.0625 * sin(x))) * fma(-0.0625, y, sin(x))), 2.0) / t_0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(cos(x) * Float64(sqrt(5.0) - 1.0))), 1.0)
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) - cos(y)))
	tmp = 0.0
	if ((y <= -0.065) || !(y <= 0.23))
		tmp = Float64(Float64(fma(t_1, Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / t_0) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(t_1, Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * fma(-0.0625, y, sin(x))), 2.0) / t_0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.065], N[Not[LessEqual[y, 0.23]], $MachinePrecision]], N[(N[(N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(t$95$1 * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.065000000000000002 or 0.23000000000000001 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.4%

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

   if -0.065000000000000002 < y < 0.23000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.0%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.065 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.065 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_0, \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, \cos x \cdot t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos y, t\_1 \cdot \cos x\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (or (<= y -0.065) (not (<= y 0.23)))
     (*
      (/
       (fma (* (sqrt 2.0) t_0) (* (sin y) (- (sin x) (* 0.0625 (sin y)))) 2.0)
       (fma 0.5 (fma (cos y) t_2 (* (cos x) t_1)) 1.0))
      0.3333333333333333)
     (*
      (fma
       t_0
       (* (fma -0.0625 (sin x) (sin y)) (* (fma -0.0625 y (sin x)) (sqrt 2.0)))
       2.0)
      (/
       0.3333333333333333
       (fma (fma t_2 (cos y) (* t_1 (cos x))) 0.5 1.0))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if ((y <= -0.065) || !(y <= 0.23)) {
		tmp = (fma((sqrt(2.0) * t_0), (sin(y) * (sin(x) - (0.0625 * sin(y)))), 2.0) / fma(0.5, fma(cos(y), t_2, (cos(x) * t_1)), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(t_0, (fma(-0.0625, sin(x), sin(y)) * (fma(-0.0625, y, sin(x)) * sqrt(2.0))), 2.0) * (0.3333333333333333 / fma(fma(t_2, cos(y), (t_1 * cos(x))), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((y <= -0.065) || !(y <= 0.23))
		tmp = Float64(Float64(fma(Float64(sqrt(2.0) * t_0), Float64(sin(y) * Float64(sin(x) - Float64(0.0625 * sin(y)))), 2.0) / fma(0.5, fma(cos(y), t_2, Float64(cos(x) * t_1)), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(t_0, Float64(fma(-0.0625, sin(x), sin(y)) * Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0))), 2.0) * Float64(0.3333333333333333 / fma(fma(t_2, cos(y), Float64(t_1 * cos(x))), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.065], N[Not[LessEqual[y, 0.23]], $MachinePrecision]], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(t$95$0 * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.065000000000000002 or 0.23000000000000001 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 56.4%

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

   if -0.065000000000000002 < y < 0.23000000000000001

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sin.f64 98.9

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

   11. Applied rewrites 98.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.065 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \sin y \cdot \left(\sin x - 0.0625 \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.021\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sin x, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.145) (not (<= x 0.021)))
     (*
      (/
       (fma
        (* (sqrt 2.0) (- (cos x) (cos y)))
        (* (- (sin y) (* 0.0625 (sin x))) (sin x))
        2.0)
       (fma 0.5 (fma (cos y) t_1 (* (cos x) t_0)) 1.0))
      0.3333333333333333)
     (/
      (+
       2.0
       (*
        (*
         (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
         (- (sin y) (/ (sin x) 16.0)))
        (fma (* (fma 0.041666666666666664 (* x x) -0.5) x) x (- 1.0 (cos y)))))
      (fma
       (* (fma -0.75 (sqrt 5.0) 0.75) x)
       x
       (fma 1.5 (fma t_1 (cos y) t_0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.145) || !(x <= 0.021)) {
		tmp = (fma((sqrt(2.0) * (cos(x) - cos(y))), ((sin(y) - (0.0625 * sin(x))) * sin(x)), 2.0) / fma(0.5, fma(cos(y), t_1, (cos(x) * t_0)), 1.0)) * 0.3333333333333333;
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * fma((fma(0.041666666666666664, (x * x), -0.5) * x), x, (1.0 - cos(y))))) / fma((fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_1, cos(y), t_0), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.145) || !(x <= 0.021))
		tmp = Float64(Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(y) - Float64(0.0625 * sin(x))) * sin(x)), 2.0) / fma(0.5, fma(cos(y), t_1, Float64(cos(x) * t_0)), 1.0)) * 0.3333333333333333);
	else
		tmp = 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))) * fma(Float64(fma(0.041666666666666664, Float64(x * x), -0.5) * x), x, Float64(1.0 - cos(y))))) / fma(Float64(fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_1, cos(y), t_0), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.145], N[Not[LessEqual[x, 0.021]], $MachinePrecision]], N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 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[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.14499999999999999 or 0.0210000000000000013 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 60.2%

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

   if -0.14499999999999999 < x < 0.0210000000000000013

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot x} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x, x, 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x}, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)} \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right) \cdot x, x, \color{blue}{1 - \cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      14. lower-cos.f64 98.8

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.021\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \sin x, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.021\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0 \cdot \cos x\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.145) (not (<= x 0.021)))
     (*
      (fma
       (- (cos x) (cos y))
       (* (fma -0.0625 (sin x) (sin y)) (* (sin x) (sqrt 2.0)))
       2.0)
      (/ 0.3333333333333333 (fma (fma t_1 (cos y) (* t_0 (cos x))) 0.5 1.0)))
     (/
      (+
       2.0
       (*
        (*
         (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
         (- (sin y) (/ (sin x) 16.0)))
        (fma (* (fma 0.041666666666666664 (* x x) -0.5) x) x (- 1.0 (cos y)))))
      (fma
       (* (fma -0.75 (sqrt 5.0) 0.75) x)
       x
       (fma 1.5 (fma t_1 (cos y) t_0) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.145) || !(x <= 0.021)) {
		tmp = fma((cos(x) - cos(y)), (fma(-0.0625, sin(x), sin(y)) * (sin(x) * sqrt(2.0))), 2.0) * (0.3333333333333333 / fma(fma(t_1, cos(y), (t_0 * cos(x))), 0.5, 1.0));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * fma((fma(0.041666666666666664, (x * x), -0.5) * x), x, (1.0 - cos(y))))) / fma((fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_1, cos(y), t_0), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.145) || !(x <= 0.021))
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(sin(x) * sqrt(2.0))), 2.0) * Float64(0.3333333333333333 / fma(fma(t_1, cos(y), Float64(t_0 * cos(x))), 0.5, 1.0)));
	else
		tmp = 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))) * fma(Float64(fma(0.041666666666666664, Float64(x * x), -0.5) * x), x, Float64(1.0 - cos(y))))) / fma(Float64(fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_1, cos(y), t_0), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.145], N[Not[LessEqual[x, 0.021]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.14499999999999999 or 0.0210000000000000013 < x

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in y around 0

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

       1. lower-sin.f64 60.2

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

   11. Applied rewrites 60.2%

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

   if -0.14499999999999999 < x < 0.0210000000000000013

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot x} + \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x, x, 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot x}, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)} \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right) \cdot x, x, \color{blue}{1 - \cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      14. lower-cos.f64 98.8

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.021\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sin x \cdot \sqrt{2}\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right) \cdot x, x, 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0116 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - 1, \mathsf{fma}\left(\sqrt{2} \cdot y, 1.00390625 \cdot \sin x, \left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.0116) (not (<= y 0.23)))
   (/
    (fma
     (fma -0.0625 (sin y) (sin x))
     (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y))
     2.0)
    (*
     3.0
     (fma
      (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
      (- 14.0 (* 3.0 (sqrt 5.0)))
      (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
   (/
    (fma
     (- (cos x) 1.0)
     (fma
      (* (sqrt 2.0) y)
      (* 1.00390625 (sin x))
      (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)))
     2.0)
    (*
     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) {
	double tmp;
	if ((y <= -0.0116) || !(y <= 0.23)) {
		tmp = fma(fma(-0.0625, sin(y), sin(x)), (((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	} else {
		tmp = fma((cos(x) - 1.0), fma((sqrt(2.0) * y), (1.00390625 * sin(x)), ((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0))), 2.0) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.0116) || !(y <= 0.23))
		tmp = Float64(fma(fma(-0.0625, sin(y), sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	else
		tmp = Float64(fma(Float64(cos(x) - 1.0), fma(Float64(sqrt(2.0) * y), Float64(1.00390625 * sin(x)), Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0))), 2.0) / 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
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.0116], N[Not[LessEqual[y, 0.23]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * y), $MachinePrecision] * N[(1.00390625 * N[Sin[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $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]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0116 or 0.23000000000000001 < y

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)} \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)} \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       8. lower-sin.f64 53.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   11. Applied rewrites 53.0%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -0.0116 < y < 0.23000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 98.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0116 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - 1, \mathsf{fma}\left(\sqrt{2} \cdot y, 1.00390625 \cdot \sin x, \left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)\\ t_1 := \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\\ \mathbf{if}\;y \leq -0.0122 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (fma
           (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
           (- 14.0 (* 3.0 (sqrt 5.0)))
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
        (t_1 (fma -0.0625 (sin y) (sin x))))
   (if (or (<= y -0.0122) (not (<= y 0.23)))
     (/ (fma t_1 (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y)) 2.0) t_0)
     (/
      (fma t_1 (* (* (- (cos x) 1.0) (sqrt 2.0)) (fma -0.0625 (sin x) y)) 2.0)
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	double t_1 = fma(-0.0625, sin(y), sin(x));
	double tmp;
	if ((y <= -0.0122) || !(y <= 0.23)) {
		tmp = fma(t_1, (((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), 2.0) / t_0;
	} else {
		tmp = fma(t_1, (((cos(x) - 1.0) * sqrt(2.0)) * fma(-0.0625, sin(x), y)), 2.0) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)))
	t_1 = fma(-0.0625, sin(y), sin(x))
	tmp = 0.0
	if ((y <= -0.0122) || !(y <= 0.23))
		tmp = Float64(fma(t_1, Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), 2.0) / t_0);
	else
		tmp = Float64(fma(t_1, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * fma(-0.0625, sin(x), y)), 2.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.0122], N[Not[LessEqual[y, 0.23]], $MachinePrecision]], N[(N[(t$95$1 * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(t$95$1 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0122000000000000008 or 0.23000000000000001 < y

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)} \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)} \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       8. lower-sin.f64 53.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   11. Applied rewrites 53.0%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -0.0122000000000000008 < y < 0.23000000000000001

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\frac{-1}{16} \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + y \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\frac{-1}{16} \cdot \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + y \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       2. distribute-rgt-out N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot \left(\frac{-1}{16} \cdot \sin x + y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot \left(\frac{-1}{16} \cdot \sin x + y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       6. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       7. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\frac{-1}{16} \cdot \sin x + y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \sin x, y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       10. lower-sin.f64 97.9

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin x}, y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   11. Applied rewrites 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0122 \lor \neg \left(y \leq 0.23\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 0.00315\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(t\_0, \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (or (<= y -0.00047) (not (<= y 0.00315)))
     (/
      (fma
       (fma -0.0625 (sin y) (sin x))
       (* (* (- 1.0 (cos y)) (sqrt 2.0)) (sin y))
       2.0)
      (*
       3.0
       (fma
        (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
        (- 14.0 (* 3.0 (sqrt 5.0)))
        (fma t_0 (cos x) 1.0))))
     (/
      (+
       2.0
       (*
        (*
         (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
         (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) 1.0)))
      (fma (fma t_0 (cos x) (/ 2.0 (+ (sqrt 5.0) 3.0))) 3.0 3.0)))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if ((y <= -0.00047) || !(y <= 0.00315)) {
		tmp = fma(fma(-0.0625, sin(y), sin(x)), (((1.0 - cos(y)) * sqrt(2.0)) * sin(y)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(t_0, cos(x), 1.0)));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / fma(fma(t_0, cos(x), (2.0 / (sqrt(5.0) + 3.0))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if ((y <= -0.00047) || !(y <= 0.00315))
		tmp = Float64(fma(fma(-0.0625, sin(y), sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * sin(y)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(t_0, cos(x), 1.0))));
	else
		tmp = 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) - 1.0))) / fma(fma(t_0, cos(x), Float64(2.0 / Float64(sqrt(5.0) + 3.0))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.00047], N[Not[LessEqual[y, 0.00315]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.69999999999999986e-4 or 0.00315 < y

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)} \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)} \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       7. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

       8. lower-sin.f64 52.8

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   11. Applied rewrites 52.8%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -4.69999999999999986e-4 < y < 0.00315

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 98.5

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

   10. Applied rewrites 98.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 0.00315\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \sin y, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := -0.0625 \cdot {\sin x}^{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.145:\\ \;\;\;\;\frac{2 + \left(t\_2 \cdot \sqrt{2}\right) \cdot t\_0}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, t\_3 \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 0.0058:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_3, \cos y, t\_1\right), 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* -0.0625 (pow (sin x) 2.0)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.145)
     (/
      (+ 2.0 (* (* t_2 (sqrt 2.0)) t_0))
      (* 3.0 (fma (fma (cos x) t_1 (* t_3 (cos y))) 0.5 1.0)))
     (if (<= x 0.0058)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (fma -0.0625 (sin y) x))
           (- (sin y) (/ (sin x) 16.0)))
          t_0))
        (fma
         (* (fma -0.75 (sqrt 5.0) 0.75) x)
         x
         (fma 1.5 (fma t_3 (cos y) t_1) 3.0)))
       (/
        (fma t_2 (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
        (*
         3.0
         (fma
          (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
          (- 14.0 (* 3.0 (sqrt 5.0)))
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = -0.0625 * pow(sin(x), 2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.145) {
		tmp = (2.0 + ((t_2 * sqrt(2.0)) * t_0)) / (3.0 * fma(fma(cos(x), t_1, (t_3 * cos(y))), 0.5, 1.0));
	} else if (x <= 0.0058) {
		tmp = (2.0 + (((sqrt(2.0) * fma(-0.0625, sin(y), x)) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma((fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_3, cos(y), t_1), 3.0));
	} else {
		tmp = fma(t_2, ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(-0.0625 * (sin(x) ^ 2.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.145)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * sqrt(2.0)) * t_0)) / Float64(3.0 * fma(fma(cos(x), t_1, Float64(t_3 * cos(y))), 0.5, 1.0)));
	elseif (x <= 0.0058)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(y), x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(Float64(fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_3, cos(y), t_1), 3.0)));
	else
		tmp = Float64(fma(t_2, Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.145], N[(N[(2.0 + N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0058], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.14499999999999999

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-cos.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 58.4

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

   8. Applied rewrites 58.4%

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

   if -0.14499999999999999 < x < 0.0058

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 98.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   if 0.0058 < x

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 54.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 54.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{5} + 3}\\ t_1 := -0.0625 \cdot {\sin y}^{2}\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;y \leq -0.00047:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(t\_2, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 0.00315:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_0\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_1 \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_0 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ (sqrt 5.0) 3.0)))
        (t_1 (* -0.0625 (pow (sin y) 2.0)))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= y -0.00047)
     (/
      (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (*
       3.0
       (fma
        (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
        (- 14.0 (* 3.0 (sqrt 5.0)))
        (fma t_2 (cos x) 1.0))))
     (if (<= y 0.00315)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ (sin x) 16.0)))
          (- (cos x) 1.0)))
        (fma (fma t_2 (cos x) t_0) 3.0 3.0))
       (/
        (+ 2.0 (* (* t_1 (sqrt 2.0)) (- (cos x) (cos y))))
        (*
         3.0
         (+
          (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
          (* t_0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (sqrt(5.0) + 3.0);
	double t_1 = -0.0625 * pow(sin(y), 2.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (y <= -0.00047) {
		tmp = fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(t_2, cos(x), 1.0)));
	} else if (y <= 0.00315) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - 1.0))) / fma(fma(t_2, cos(x), t_0), 3.0, 3.0);
	} else {
		tmp = (2.0 + ((t_1 * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (t_0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 / Float64(sqrt(5.0) + 3.0))
	t_1 = Float64(-0.0625 * (sin(y) ^ 2.0))
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (y <= -0.00047)
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(t_2, cos(x), 1.0))));
	elseif (y <= 0.00315)
		tmp = 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) - 1.0))) / fma(fma(t_2, cos(x), t_0), 3.0, 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * sqrt(2.0)) * 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(t_0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[y, -0.00047], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00315], 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] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$1 * N[Sqrt[2.0], $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[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.69999999999999986e-4

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -4.69999999999999986e-4 < y < 0.00315

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 98.5

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

   10. Applied rewrites 98.5%

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

   if 0.00315 < y

   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. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 54.5

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

   7. Applied rewrites 54.5%

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

Alternative 19: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{2}{\sqrt{5} + 3}\\ t_2 := -0.0625 \cdot {\sin y}^{2}\\ t_3 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;y \leq -0.00047:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(t\_3, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 0.00315:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, \cos x, t\_1\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_2 \cdot \sqrt{2}\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_1 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (/ 2.0 (+ (sqrt 5.0) 3.0)))
        (t_2 (* -0.0625 (pow (sin y) 2.0)))
        (t_3 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= y -0.00047)
     (/
      (fma t_2 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (*
       3.0
       (fma
        (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
        (- 14.0 (* 3.0 (sqrt 5.0)))
        (fma t_3 (cos x) 1.0))))
     (if (<= y 0.00315)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (fma -0.0625 y (sin x)))
           (- (sin y) (/ (sin x) 16.0)))
          t_0))
        (fma (fma t_3 (cos x) t_1) 3.0 3.0))
       (/
        (+ 2.0 (* (* t_2 (sqrt 2.0)) t_0))
        (*
         3.0
         (+
          (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
          (* t_1 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 2.0 / (sqrt(5.0) + 3.0);
	double t_2 = -0.0625 * pow(sin(y), 2.0);
	double t_3 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (y <= -0.00047) {
		tmp = fma(t_2, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(t_3, cos(x), 1.0)));
	} else if (y <= 0.00315) {
		tmp = (2.0 + (((sqrt(2.0) * fma(-0.0625, y, sin(x))) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(fma(t_3, cos(x), t_1), 3.0, 3.0);
	} else {
		tmp = (2.0 + ((t_2 * sqrt(2.0)) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (t_1 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(2.0 / Float64(sqrt(5.0) + 3.0))
	t_2 = Float64(-0.0625 * (sin(y) ^ 2.0))
	t_3 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (y <= -0.00047)
		tmp = Float64(fma(t_2, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(t_3, cos(x), 1.0))));
	elseif (y <= 0.00315)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(-0.0625, y, sin(x))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(fma(t_3, cos(x), t_1), 3.0, 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * sqrt(2.0)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(t_1 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[y, -0.00047], N[(N[(t$95$2 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00315], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $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[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.69999999999999986e-4

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -4.69999999999999986e-4 < y < 0.00315

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 98.5

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

   10. Applied rewrites 98.5%

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

   if 0.00315 < y

   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. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 54.5

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

   7. Applied rewrites 54.5%

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

Alternative 20: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{2}{\sqrt{5} + 3}\\ t_2 := -0.0625 \cdot {\sin y}^{2}\\ t_3 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;y \leq -0.00047:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(t\_3, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 0.00315:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, \cos x, t\_1\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_2 \cdot \sqrt{2}\right) \cdot t\_0}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + t\_1 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (/ 2.0 (+ (sqrt 5.0) 3.0)))
        (t_2 (* -0.0625 (pow (sin y) 2.0)))
        (t_3 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= y -0.00047)
     (/
      (fma t_2 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (*
       3.0
       (fma
        (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
        (- 14.0 (* 3.0 (sqrt 5.0)))
        (fma t_3 (cos x) 1.0))))
     (if (<= y 0.00315)
       (/
        (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0))
        (fma (fma t_3 (cos x) t_1) 3.0 3.0))
       (/
        (+ 2.0 (* (* t_2 (sqrt 2.0)) t_0))
        (*
         3.0
         (+
          (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x)))
          (* t_1 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 2.0 / (sqrt(5.0) + 3.0);
	double t_2 = -0.0625 * pow(sin(y), 2.0);
	double t_3 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (y <= -0.00047) {
		tmp = fma(t_2, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(t_3, cos(x), 1.0)));
	} else if (y <= 0.00315) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(fma(t_3, cos(x), t_1), 3.0, 3.0);
	} else {
		tmp = (2.0 + ((t_2 * sqrt(2.0)) * t_0)) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (t_1 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(2.0 / Float64(sqrt(5.0) + 3.0))
	t_2 = Float64(-0.0625 * (sin(y) ^ 2.0))
	t_3 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (y <= -0.00047)
		tmp = Float64(fma(t_2, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(t_3, cos(x), 1.0))));
	elseif (y <= 0.00315)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(fma(t_3, cos(x), t_1), 3.0, 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * sqrt(2.0)) * t_0)) / Float64(3.0 * Float64(Float64(1.0 + Float64(Float64(Float64(sqrt(5.0) - 1.0) / 2.0) * cos(x))) + Float64(t_1 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[y, -0.00047], N[(N[(t$95$2 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00315], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$3 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $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[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.69999999999999986e-4

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -4.69999999999999986e-4 < y < 0.00315

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 98.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin x} \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-sqrt.f64 98.3

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 98.3%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   if 0.00315 < y

   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. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.0

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 54.5

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 54.5%

      \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := -0.0625 \cdot {\sin y}^{2}\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;y \leq -0.00047:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(t\_2, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 0.00315:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_1 \cdot \sqrt{2}\right) \cdot t\_0}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* -0.0625 (pow (sin y) 2.0)))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= y -0.00047)
     (/
      (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (*
       3.0
       (fma
        (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
        (- 14.0 (* 3.0 (sqrt 5.0)))
        (fma t_2 (cos x) 1.0))))
     (if (<= y 0.00315)
       (/
        (+ 2.0 (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0))
        (fma (fma t_2 (cos x) (/ 2.0 (+ (sqrt 5.0) 3.0))) 3.0 3.0))
       (/
        (+ 2.0 (* (* t_1 (sqrt 2.0)) t_0))
        (*
         3.0
         (fma
          (fma (cos x) (- (sqrt 5.0) 1.0) (* (- 3.0 (sqrt 5.0)) (cos y)))
          0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = -0.0625 * pow(sin(y), 2.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (y <= -0.00047) {
		tmp = fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(t_2, cos(x), 1.0)));
	} else if (y <= 0.00315) {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0)) / fma(fma(t_2, cos(x), (2.0 / (sqrt(5.0) + 3.0))), 3.0, 3.0);
	} else {
		tmp = (2.0 + ((t_1 * sqrt(2.0)) * t_0)) / (3.0 * fma(fma(cos(x), (sqrt(5.0) - 1.0), ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(-0.0625 * (sin(y) ^ 2.0))
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (y <= -0.00047)
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(t_2, cos(x), 1.0))));
	elseif (y <= 0.00315)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0)) / fma(fma(t_2, cos(x), Float64(2.0 / Float64(sqrt(5.0) + 3.0))), 3.0, 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * sqrt(2.0)) * t_0)) / Float64(3.0 * fma(fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[y, -0.00047], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00315], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.69999999999999986e-4

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -4.69999999999999986e-4 < y < 0.00315

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 98.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin x} \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-sqrt.f64 98.3

         \[\leadsto \frac{2 + \left(\left(\sin x \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 98.3%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   if 0.00315 < y

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-sqrt.f64 54.4

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 54.4%

      \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)\\ t_1 := -0.0625 \cdot {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.00116:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{elif}\;y \leq 0.0132:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_1 \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (fma
           (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
           (- 14.0 (* 3.0 (sqrt 5.0)))
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
        (t_1 (* -0.0625 (pow (sin y) 2.0))))
   (if (<= y -0.00116)
     (/ (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0) t_0)
     (if (<= y 0.0132)
       (/
        (fma (* -0.0625 (pow (sin x) 2.0)) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
        t_0)
       (/
        (+ 2.0 (* (* t_1 (sqrt 2.0)) (- (cos x) (cos y))))
        (*
         3.0
         (fma
          (fma (cos x) (- (sqrt 5.0) 1.0) (* (- 3.0 (sqrt 5.0)) (cos y)))
          0.5
          1.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	double t_1 = -0.0625 * pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.00116) {
		tmp = fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0;
	} else if (y <= 0.0132) {
		tmp = fma((-0.0625 * pow(sin(x), 2.0)), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_0;
	} else {
		tmp = (2.0 + ((t_1 * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * fma(fma(cos(x), (sqrt(5.0) - 1.0), ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)))
	t_1 = Float64(-0.0625 * (sin(y) ^ 2.0))
	tmp = 0.0
	if (y <= -0.00116)
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0);
	elseif (y <= 0.0132)
		tmp = Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * fma(fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00116], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 0.0132], N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00116

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -0.00116 < y < 0.0132

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 97.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 97.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if 0.0132 < y

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-sqrt.f64 54.5

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 54.5%

      \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := -0.0625 \cdot {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.145:\\ \;\;\;\;\frac{2 + \left(t\_1 \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 29000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(t\_0 \cdot \cos x, 0.5, 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (* -0.0625 (pow (sin x) 2.0))))
   (if (<= x -0.145)
     (/
      (+ 2.0 (* (* t_1 (sqrt 2.0)) (- (cos x) (cos y))))
      (* 3.0 (fma (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 0.5 1.0)))
     (if (<= x 29000000000000.0)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (*
         3.0
         (+
          (fma (* t_0 (cos x)) 0.5 1.0)
          (* (/ 2.0 (+ (sqrt 5.0) 3.0)) (cos y)))))
       (/
        (fma t_1 (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
        (*
         3.0
         (fma
          (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
          (- 14.0 (* 3.0 (sqrt 5.0)))
          (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = -0.0625 * pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.145) {
		tmp = (2.0 + ((t_1 * sqrt(2.0)) * (cos(x) - cos(y)))) / (3.0 * fma(fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
	} else if (x <= 29000000000000.0) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * (fma((t_0 * cos(x)), 0.5, 1.0) + ((2.0 / (sqrt(5.0) + 3.0)) * cos(y))));
	} else {
		tmp = fma(t_1, ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(-0.0625 * (sin(x) ^ 2.0))
	tmp = 0.0
	if (x <= -0.145)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / Float64(3.0 * fma(fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)));
	elseif (x <= 29000000000000.0)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * Float64(fma(Float64(t_0 * cos(x)), 0.5, 1.0) + Float64(Float64(2.0 / Float64(sqrt(5.0) + 3.0)) * cos(y)))));
	else
		tmp = Float64(fma(t_1, Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	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[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.145], N[(N[(2.0 + N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 29000000000000.0], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.14499999999999999

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-sqrt.f64 58.4

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 58.4%

      \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   if -0.14499999999999999 < x < 2.9e13

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 95.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 95.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\frac{\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}}{2} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\frac{\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}}{2} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2}} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\frac{1}{2}} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-fma.f64 95.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right), 0.5, 1\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}, \frac{1}{2}, 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}, \frac{1}{2}, 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      13. lower-*.f64 95.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}, 0.5, 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. Applied rewrites 95.2%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x, 0.5, 1\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   if 2.9e13 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   6. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 55.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 55.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)\\ \mathbf{if}\;y \leq -0.00116 \lor \neg \left(y \leq 0.0132\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (fma
           (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
           (- 14.0 (* 3.0 (sqrt 5.0)))
           (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))))
   (if (or (<= y -0.00116) (not (<= y 0.0132)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      t_0)
     (/
      (fma (* -0.0625 (pow (sin x) 2.0)) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	double tmp;
	if ((y <= -0.00116) || !(y <= 0.0132)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0;
	} else {
		tmp = fma((-0.0625 * pow(sin(x), 2.0)), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)))
	tmp = 0.0
	if ((y <= -0.00116) || !(y <= 0.0132))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0);
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.00116], N[Not[LessEqual[y, 0.0132]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00116 or 0.0132 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   6. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 52.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 52.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -0.00116 < y < 0.0132

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 97.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 97.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00116 \lor \neg \left(y \leq 0.0132\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;y \leq -0.0012 \lor \neg \left(y \leq 0.0132\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
   (if (or (<= y -0.0012) (not (<= y 0.0132)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* 3.0 (fma (/ (cos y) (+ (sqrt 5.0) 3.0)) 2.0 t_0)))
     (/
      (fma (* -0.0625 (pow (sin x) 2.0)) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      (*
       3.0
       (fma
        (/ (* (cos y) 2.0) (fma (sqrt 5.0) 5.0 27.0))
        (- 14.0 (* 3.0 (sqrt 5.0)))
        t_0))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double tmp;
	if ((y <= -0.0012) || !(y <= 0.0132)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma((cos(y) / (sqrt(5.0) + 3.0)), 2.0, t_0));
	} else {
		tmp = fma((-0.0625 * pow(sin(x), 2.0)), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (3.0 * fma(((cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), (14.0 - (3.0 * sqrt(5.0))), t_0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	tmp = 0.0
	if ((y <= -0.0012) || !(y <= 0.0132))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), 2.0, t_0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(Float64(cos(y) * 2.0) / fma(sqrt(5.0), 5.0, 27.0)), Float64(14.0 - Float64(3.0 * sqrt(5.0))), t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[y, -0.0012], N[Not[LessEqual[y, 0.0132]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[(N[Cos[y], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] * 5.0 + 27.0), $MachinePrecision]), $MachinePrecision] * N[(14.0 - N[(3.0 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00119999999999999989 or 0.0132 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 52.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 52.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5}} + 3}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1}\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + 1\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x} + 1\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)}\right)} \]

   10. Applied rewrites 52.1%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if -0.00119999999999999989 < y < 0.0132

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      11. lower-sqrt.f64 97.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 97.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0012 \lor \neg \left(y \leq 0.0132\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y \cdot 2}{\mathsf{fma}\left(\sqrt{5}, 5, 27\right)}, 14 - 3 \cdot \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + 3\\ t_1 := \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_3 := \frac{2}{t\_0}\\ \mathbf{if}\;y \leq -0.00047:\\ \;\;\;\;\frac{t\_1}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{t\_0}, 2, \mathsf{fma}\left(t\_2, \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, t\_3\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x, 0.5, 1\right) + t\_3 \cdot \cos y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) 3.0))
        (t_1
         (fma
          (* -0.0625 (pow (sin y) 2.0))
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5))
        (t_3 (/ 2.0 t_0)))
   (if (<= y -0.00047)
     (/ t_1 (* 3.0 (fma (/ (cos y) t_0) 2.0 (fma t_2 (cos x) 1.0))))
     (if (<= y 3.25e-6)
       (/
        (+
         2.0
         (* (* (* -0.0625 (pow (sin x) 2.0)) (sqrt 2.0)) (- (cos x) (cos y))))
        (fma (fma t_2 (cos x) t_3) 3.0 3.0))
       (/
        t_1
        (*
         3.0
         (+ (fma (* (- (sqrt 5.0) 1.0) (cos x)) 0.5 1.0) (* t_3 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + 3.0;
	double t_1 = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double t_3 = 2.0 / t_0;
	double tmp;
	if (y <= -0.00047) {
		tmp = t_1 / (3.0 * fma((cos(y) / t_0), 2.0, fma(t_2, cos(x), 1.0)));
	} else if (y <= 3.25e-6) {
		tmp = (2.0 + (((-0.0625 * pow(sin(x), 2.0)) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(fma(t_2, cos(x), t_3), 3.0, 3.0);
	} else {
		tmp = t_1 / (3.0 * (fma(((sqrt(5.0) - 1.0) * cos(x)), 0.5, 1.0) + (t_3 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + 3.0)
	t_1 = fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	t_3 = Float64(2.0 / t_0)
	tmp = 0.0
	if (y <= -0.00047)
		tmp = Float64(t_1 / Float64(3.0 * fma(Float64(cos(y) / t_0), 2.0, fma(t_2, cos(x), 1.0))));
	elseif (y <= 3.25e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(fma(t_2, cos(x), t_3), 3.0, 3.0));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(fma(Float64(Float64(sqrt(5.0) - 1.0) * cos(x)), 0.5, 1.0) + Float64(t_3 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.00047], N[(t$95$1 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision] * 2.0 + N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.25e-6], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + N[(t$95$3 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.69999999999999986e-4

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 50.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5}} + 3}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1}\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + 1\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x} + 1\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)}\right)} \]

   10. Applied rewrites 50.0%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if -4.69999999999999986e-4 < y < 3.2499999999999998e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      6. lower-sqrt.f64 98.9

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   if 3.2499999999999998e-6 < y

   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. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 54.3

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 54.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\frac{\left(\sqrt{5} - 1\right) \cdot \cos x}{2}} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\frac{\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}}{2} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\frac{\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}}{2} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \frac{1}{2}} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\frac{1}{2}} + 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-fma.f64 54.3

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right), 0.5, 1\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)}, \frac{1}{2}, 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}, \frac{1}{2}, 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      13. lower-*.f64 54.3

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}, 0.5, 1\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. Applied rewrites 54.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\left(\sqrt{5} - 1\right) \cdot \cos x, 0.5, 1\right)} + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + 3\\ t_1 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 3.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{t\_0}, 2, \mathsf{fma}\left(t\_1, \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, \frac{2}{t\_0}\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) 3.0)) (t_1 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (or (<= y -0.00047) (not (<= y 3.25e-6)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* 3.0 (fma (/ (cos y) t_0) 2.0 (fma t_1 (cos x) 1.0))))
     (/
      (+
       2.0
       (* (* (* -0.0625 (pow (sin x) 2.0)) (sqrt 2.0)) (- (cos x) (cos y))))
      (fma (fma t_1 (cos x) (/ 2.0 t_0)) 3.0 3.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + 3.0;
	double t_1 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if ((y <= -0.00047) || !(y <= 3.25e-6)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma((cos(y) / t_0), 2.0, fma(t_1, cos(x), 1.0)));
	} else {
		tmp = (2.0 + (((-0.0625 * pow(sin(x), 2.0)) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(fma(t_1, cos(x), (2.0 / t_0)), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + 3.0)
	t_1 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if ((y <= -0.00047) || !(y <= 3.25e-6))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(cos(y) / t_0), 2.0, fma(t_1, cos(x), 1.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(fma(t_1, cos(x), Float64(2.0 / t_0)), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.00047], N[Not[LessEqual[y, 3.25e-6]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision] * 2.0 + N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.69999999999999986e-4 or 3.2499999999999998e-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. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 52.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 52.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5}} + 3}, 2, 1 + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 1}\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + 1\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x} + 1\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right), \cos x, 1\right)}\right)} \]

   10. Applied rewrites 52.1%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

   if -4.69999999999999986e-4 < y < 3.2499999999999998e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      6. lower-sqrt.f64 98.9

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 3.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 3.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.00047) (not (<= y 3.25e-6)))
   (/
    (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
    (*
     3.0
     (fma
      (fma (cos x) (- (sqrt 5.0) 1.0) (* (- 3.0 (sqrt 5.0)) (cos y)))
      0.5
      1.0)))
   (/
    (+
     2.0
     (* (* (* -0.0625 (pow (sin x) 2.0)) (sqrt 2.0)) (- (cos x) (cos y))))
    (fma
     (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (/ 2.0 (+ (sqrt 5.0) 3.0)))
     3.0
     3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.00047) || !(y <= 3.25e-6)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(fma(cos(x), (sqrt(5.0) - 1.0), ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
	} else {
		tmp = (2.0 + (((-0.0625 * pow(sin(x), 2.0)) * sqrt(2.0)) * (cos(x) - cos(y)))) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (2.0 / (sqrt(5.0) + 3.0))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.00047) || !(y <= 3.25e-6))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * sqrt(2.0)) * Float64(cos(x) - cos(y)))) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(2.0 / Float64(sqrt(5.0) + 3.0))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.00047], N[Not[LessEqual[y, 3.25e-6]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.69999999999999986e-4 or 3.2499999999999998e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      11. lower-sqrt.f64 52.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 52.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   if -4.69999999999999986e-4 < y < 3.2499999999999998e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      6. lower-sqrt.f64 98.9

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 3.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 3.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.00047) (not (<= y 3.25e-6)))
   (/
    (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
    (*
     3.0
     (fma
      (fma (cos x) (- (sqrt 5.0) 1.0) (* (- 3.0 (sqrt 5.0)) (cos y)))
      0.5
      1.0)))
   (/
    (fma (* -0.0625 (pow (sin x) 2.0)) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
    (fma
     (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (/ 2.0 (+ (sqrt 5.0) 3.0)))
     3.0
     3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.00047) || !(y <= 3.25e-6)) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(fma(cos(x), (sqrt(5.0) - 1.0), ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
	} else {
		tmp = fma((-0.0625 * pow(sin(x), 2.0)), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (2.0 / (sqrt(5.0) + 3.0))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.00047) || !(y <= 3.25e-6))
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(x) ^ 2.0)), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(2.0 / Float64(sqrt(5.0) + 3.0))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.00047], N[Not[LessEqual[y, 3.25e-6]], $MachinePrecision]], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.69999999999999986e-4 or 3.2499999999999998e-6 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      11. lower-sqrt.f64 52.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 52.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   if -4.69999999999999986e-4 < y < 3.2499999999999998e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      11. lower-sqrt.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00047 \lor \neg \left(y \leq 3.25 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.145:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, t\_1 \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.145)
     (/
      (fma
       (* (- 0.5 (* 0.5 (cos (+ x x)))) (sqrt 2.0))
       (fma -0.0625 (cos x) 0.0625)
       2.0)
      (* 3.0 (fma (fma (cos x) t_0 (* t_1 (cos y))) 0.5 1.0)))
     (if (<= x 0.00125)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (*
         3.0
         (fma
          t_0
          (fma -0.25 (* x x) 0.5)
          (fma (/ (cos y) (+ (sqrt 5.0) 3.0)) 2.0 1.0))))
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma (cos x) -0.0625 0.0625) 2.0)
        (fma 1.5 (fma (cos y) t_1 (* (cos x) t_0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.145) {
		tmp = fma(((0.5 - (0.5 * cos((x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (3.0 * fma(fma(cos(x), t_0, (t_1 * cos(y))), 0.5, 1.0));
	} else if (x <= 0.00125) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(t_0, fma(-0.25, (x * x), 0.5), fma((cos(y) / (sqrt(5.0) + 3.0)), 2.0, 1.0)));
	} else {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(y), t_1, (cos(x) * t_0)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.145)
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(3.0 * fma(fma(cos(x), t_0, Float64(t_1 * cos(y))), 0.5, 1.0)));
	elseif (x <= 0.00125)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(t_0, fma(-0.25, Float64(x * x), 0.5), fma(Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), 2.0, 1.0))));
	else
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(y), t_1, Float64(cos(x) * t_0)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.145], N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00125], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.14499999999999999

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      16. lower-cos.f64 58.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.2%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   if -0.14499999999999999 < x < 0.00125000000000000003

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 97.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 97.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right) + 1\right)}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\left(\frac{-1}{4} \cdot {x}^{2}\right) \cdot \left(\sqrt{5} - 1\right)} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)} + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} - 1}, \frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5}} - 1, \frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {x}^{2}, \frac{1}{2}\right)}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{x \cdot x}, \frac{1}{2}\right), 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{x \cdot x}, \frac{1}{2}\right), 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4}, x \cdot x, \frac{1}{2}\right), \color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + 1\right)} \]

   10. Applied rewrites 97.0%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, 1\right)\right)}} \]

   if 0.00125000000000000003 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 54.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 54.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 54.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 3\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 31: 79.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.00125\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -0.145) (not (<= x 0.00125)))
     (/
      (fma
       (* (- 0.5 (* 0.5 (cos (+ x x)))) (sqrt 2.0))
       (fma -0.0625 (cos x) 0.0625)
       2.0)
      (* 3.0 (fma (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 0.5 1.0)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (*
       3.0
       (fma
        t_0
        (fma -0.25 (* x x) 0.5)
        (fma (/ (cos y) (+ (sqrt 5.0) 3.0)) 2.0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -0.145) || !(x <= 0.00125)) {
		tmp = fma(((0.5 - (0.5 * cos((x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (3.0 * fma(fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma(t_0, fma(-0.25, (x * x), 0.5), fma((cos(y) / (sqrt(5.0) + 3.0)), 2.0, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -0.145) || !(x <= 0.00125))
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(3.0 * fma(fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(t_0, fma(-0.25, Float64(x * x), 0.5), fma(Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), 2.0, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.145], N[Not[LessEqual[x, 0.00125]], $MachinePrecision]], N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.14499999999999999 or 0.00125000000000000003 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 99.0

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      16. lower-cos.f64 56.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   if -0.14499999999999999 < x < 0.00125000000000000003

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 97.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 97.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right) + 1\right)}} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\left(\frac{-1}{4} \cdot {x}^{2}\right) \cdot \left(\sqrt{5} - 1\right)} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)} + \left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5} - 1}, \frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{5}} - 1, \frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{\mathsf{fma}\left(\frac{-1}{4}, {x}^{2}, \frac{1}{2}\right)}, 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{x \cdot x}, \frac{1}{2}\right), 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4}, \color{blue}{x \cdot x}, \frac{1}{2}\right), 2 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(\frac{-1}{4}, x \cdot x, \frac{1}{2}\right), \color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + 1\right)} \]

   10. Applied rewrites 97.0%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.00125\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 79.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.00125\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, t\_0 \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos y, t\_1\right), 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -0.145) (not (<= x 0.00125)))
     (/
      (fma
       (* (- 0.5 (* 0.5 (cos (+ x x)))) (sqrt 2.0))
       (fma -0.0625 (cos x) 0.0625)
       2.0)
      (* 3.0 (fma (fma (cos x) t_1 (* t_0 (cos y))) 0.5 1.0)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (fma
       (* (fma -0.75 (sqrt 5.0) 0.75) x)
       x
       (fma 1.5 (fma t_0 (cos y) t_1) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -0.145) || !(x <= 0.00125)) {
		tmp = fma(((0.5 - (0.5 * cos((x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (3.0 * fma(fma(cos(x), t_1, (t_0 * cos(y))), 0.5, 1.0));
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_0, cos(y), t_1), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -0.145) || !(x <= 0.00125))
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(3.0 * fma(fma(cos(x), t_1, Float64(t_0 * cos(y))), 0.5, 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(fma(-0.75, sqrt(5.0), 0.75) * x), x, fma(1.5, fma(t_0, cos(y), t_1), 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[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.145], N[Not[LessEqual[x, 0.00125]], $MachinePrecision]], N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(-0.75 * N[Sqrt[5.0], $MachinePrecision] + 0.75), $MachinePrecision] * x), $MachinePrecision] * x + N[(1.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.14499999999999999 or 0.00125000000000000003 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 99.0

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      16. lower-cos.f64 56.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.1%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   if -0.14499999999999999 < x < 0.00125000000000000003

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

   5. Applied rewrites 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, \mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

      11. lower-sqrt.f64 97.0

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]

   8. Applied rewrites 97.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.145 \lor \neg \left(x \leq 0.00125\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 33: 79.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-7} \lor \neg \left(x \leq 54000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, t\_0, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)))
   (if (or (<= x -7.5e-7) (not (<= x 54000.0)))
     (/
      (fma
       (* (- 0.5 (* 0.5 (cos (+ x x)))) (sqrt 2.0))
       (fma -0.0625 (cos x) 0.0625)
       2.0)
      (* 3.0 (fma (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 0.5 1.0)))
     (/
      (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
      (* 3.0 (fma (/ (cos y) (+ (sqrt 5.0) 3.0)) 2.0 (fma 0.5 t_0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double tmp;
	if ((x <= -7.5e-7) || !(x <= 54000.0)) {
		tmp = fma(((0.5 - (0.5 * cos((x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / (3.0 * fma(fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma((cos(y) / (sqrt(5.0) + 3.0)), 2.0, fma(0.5, t_0, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if ((x <= -7.5e-7) || !(x <= 54000.0))
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / Float64(3.0 * fma(fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), 2.0, fma(0.5, t_0, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -7.5e-7], N[Not[LessEqual[x, 54000.0]], $MachinePrecision]], N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(0.5 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000002e-7 or 54000 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. lower-cos.f64 99.0

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

   5. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      13. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

      16. lower-cos.f64 56.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   8. Applied rewrites 56.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.3%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

   if -7.5000000000000002e-7 < x < 54000

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 98.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5}} + 3}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 1}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} - 1, 1\right)}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{5} - 1}, 1\right)\right)} \]

      15. lower-sqrt.f64 98.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}} - 1, 1\right)\right)} \]

   10. Applied rewrites 98.1%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-7} \lor \neg \left(x \leq 54000\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 34: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := \sqrt{5} + 3\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 54000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{t\_2}, 2, \mathsf{fma}\left(0.5, t\_0, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_1, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{t\_2}\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (+ (sqrt 5.0) 3.0)))
   (if (<= x -1.45e-6)
     (*
      (/
       (fma (* t_1 (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
       (fma 0.5 (fma (cos x) t_0 (- 3.0 (sqrt 5.0))) 1.0))
      0.3333333333333333)
     (if (<= x 54000.0)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (* 3.0 (fma (/ (cos y) t_2) 2.0 (fma 0.5 t_0 1.0))))
       (/
        (fma (* -0.0625 t_1) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
        (fma (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (/ 2.0 t_2)) 3.0 3.0))))))

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 = sqrt(5.0) + 3.0;
	double tmp;
	if (x <= -1.45e-6) {
		tmp = (fma((t_1 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), t_0, (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	} else if (x <= 54000.0) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (3.0 * fma((cos(y) / t_2), 2.0, fma(0.5, t_0, 1.0)));
	} else {
		tmp = fma((-0.0625 * t_1), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (2.0 / t_2)), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(sqrt(5.0) + 3.0)
	tmp = 0.0
	if (x <= -1.45e-6)
		tmp = Float64(Float64(fma(Float64(t_1 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), t_0, Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333);
	elseif (x <= 54000.0)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(3.0 * fma(Float64(cos(y) / t_2), 2.0, fma(0.5, t_0, 1.0))));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_1), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(2.0 / t_2)), 3.0, 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[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[x, -1.45e-6], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 54000.0], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] / t$95$2), $MachinePrecision] * 2.0 + N[(0.5 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$1), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(2.0 / t$95$2), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4500000000000001e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

   5. Applied rewrites 8.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -1.4500000000000001e-6 < x < 54000

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 98.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(2 \cdot \frac{\cos y}{3 + \sqrt{5}} + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 1\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}} \cdot 2} + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 1\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{\cos y}{3 + \sqrt{5}} \cdot 2 + \color{blue}{\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{\cos y}{3 + \sqrt{5}}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{\cos y}}{3 + \sqrt{5}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5} + 3}}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\color{blue}{\sqrt{5}} + 3}, 2, 1 + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 1}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} - 1, 1\right)}\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{5} - 1}, 1\right)\right)} \]

      15. lower-sqrt.f64 98.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}} - 1, 1\right)\right)} \]

   10. Applied rewrites 98.1%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \sqrt{5} - 1, 1\right)\right)}} \]

   if 54000 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 54.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      11. lower-sqrt.f64 53.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 53.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 35: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} + 3\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 54000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\cos y}{t\_1}, 2, t\_2\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_0, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos x, \frac{2}{t\_1}\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (+ (sqrt 5.0) 3.0))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= x -1.45e-6)
     (*
      (/
       (fma (* t_0 (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
       (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 1.0))
      0.3333333333333333)
     (if (<= x 54000.0)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (fma (/ (cos y) t_1) 2.0 t_2) 3.0 3.0))
       (/
        (fma (* -0.0625 t_0) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
        (fma (fma t_2 (cos x) (/ 2.0 t_1)) 3.0 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) + 3.0;
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (x <= -1.45e-6) {
		tmp = (fma((t_0 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
	} else if (x <= 54000.0) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma((cos(y) / t_1), 2.0, t_2), 3.0, 3.0);
	} else {
		tmp = fma((-0.0625 * t_0), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(t_2, cos(x), (2.0 / t_1)), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) + 3.0)
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (x <= -1.45e-6)
		tmp = Float64(Float64(fma(Float64(t_0 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333);
	elseif (x <= 54000.0)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(fma(Float64(cos(y) / t_1), 2.0, t_2), 3.0, 3.0));
	else
		tmp = Float64(fma(Float64(-0.0625 * t_0), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(t_2, cos(x), Float64(2.0 / t_1)), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -1.45e-6], N[(N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 54000.0], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[y], $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0 + t$95$2), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$0), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4500000000000001e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

   5. Applied rewrites 8.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -1.4500000000000001e-6 < x < 54000

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

      11. lower-sqrt.f64 98.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. Applied rewrites 98.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}, 3, 3\right)}} \]

   10. Applied rewrites 98.1%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right), 3, 3\right)}} \]

   if 54000 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 54.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      11. lower-sqrt.f64 53.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 53.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 36: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 54000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot t\_2, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -1.45e-6)
     (*
      (/
       (fma (* t_2 (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
       (fma 0.5 (fma (cos x) t_0 t_1) 1.0))
      0.3333333333333333)
     (if (<= x 54000.0)
       (*
        (/
         (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma (cos y) t_1 t_0) 1.0))
        0.3333333333333333)
       (/
        (fma (* -0.0625 t_2) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (/ 2.0 (+ (sqrt 5.0) 3.0)))
         3.0
         3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -1.45e-6) {
		tmp = (fma((t_2 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), t_0, t_1), 1.0)) * 0.3333333333333333;
	} else if (x <= 54000.0) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_1, t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma((-0.0625 * t_2), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (2.0 / (sqrt(5.0) + 3.0))), 3.0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -1.45e-6)
		tmp = Float64(Float64(fma(Float64(t_2 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), t_0, t_1), 1.0)) * 0.3333333333333333);
	elseif (x <= 54000.0)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_1, t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(Float64(-0.0625 * t_2), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(2.0 / Float64(sqrt(5.0) + 3.0))), 3.0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -1.45e-6], N[(N[(N[(N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 54000.0], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(-0.0625 * t$95$2), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4500000000000001e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

   5. Applied rewrites 8.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -1.4500000000000001e-6 < x < 54000

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

   5. Applied rewrites 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]

   if 54000 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      2. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      4. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      9. rem-square-sqrt N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

      15. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

   7. Applied rewrites 54.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

      11. lower-sqrt.f64 53.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]

   10. Applied rewrites 53.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 37: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2} \cdot \sqrt{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(\cos x, t\_0, t\_2\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, t\_3, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 54000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_2, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, t\_3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (* (pow (sin x) 2.0) (sqrt 2.0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma (cos x) t_0 t_2)))
   (if (<= x -1.45e-6)
     (*
      (/ (fma t_1 (fma -0.0625 (cos x) 0.0625) 2.0) (fma 0.5 t_3 1.0))
      0.3333333333333333)
     (if (<= x 54000.0)
       (*
        (/
         (fma (* -0.0625 (pow (sin y) 2.0)) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
         (fma 0.5 (fma (cos y) t_2 t_0) 1.0))
        0.3333333333333333)
       (/ (fma t_1 (fma (cos x) -0.0625 0.0625) 2.0) (fma 1.5 t_3 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(x), 2.0) * sqrt(2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(cos(x), t_0, t_2);
	double tmp;
	if (x <= -1.45e-6) {
		tmp = (fma(t_1, fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, t_3, 1.0)) * 0.3333333333333333;
	} else if (x <= 54000.0) {
		tmp = (fma((-0.0625 * pow(sin(y), 2.0)), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_2, t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(t_1, fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, t_3, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64((sin(x) ^ 2.0) * sqrt(2.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(cos(x), t_0, t_2)
	tmp = 0.0
	if (x <= -1.45e-6)
		tmp = Float64(Float64(fma(t_1, fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, t_3, 1.0)) * 0.3333333333333333);
	elseif (x <= 54000.0)
		tmp = Float64(Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(0.5, fma(cos(y), t_2, t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(fma(t_1, fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, t_3, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision]}, If[LessEqual[x, -1.45e-6], N[(N[(N[(t$95$1 * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 54000.0], N[(N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(t$95$1 * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$3 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4500000000000001e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

   5. Applied rewrites 8.0%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -1.4500000000000001e-6 < x < 54000

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

   5. Applied rewrites 98.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)} \cdot \frac{1}{3}} \]

   8. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \cdot 0.3333333333333333} \]

   if 54000 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 54.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 54.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      13. lower-sqrt.f64 53.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 53.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 38: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (/
   (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
   (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 1.0))
  0.3333333333333333))

C

double code(double x, double y) {
	return (fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333;
}

Julia

function code(x, y)
	return Float64(Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 1.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{-3}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\frac{-3}{4} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot {x}^{2}\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot {x}^{2}} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x\right) \cdot x} + 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right)\right) \cdot x}, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\left(\frac{-3}{4} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   8. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(\frac{-3}{4} \cdot \sqrt{5} + \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \color{blue}{\sqrt{5}}, \frac{-3}{4} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{-3}{4} \cdot \color{blue}{-1}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \color{blue}{\frac{3}{4}}\right) \cdot x, x, 3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)} \]

   13. +-commutative N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-3}{4}, \sqrt{5}, \frac{3}{4}\right) \cdot x, x, 3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}\right)} \]

5. Applied rewrites 45.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.75, \sqrt{5}, 0.75\right) \cdot x, x, \mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 3\right)\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \frac{1}{3}} \]

8. Applied rewrites 58.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
9. Add Preprocessing

Alternative 39: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma (cos x) -0.0625 0.0625) 2.0)
  (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 61.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 61.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 58.9

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 58.9%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
9. Add Preprocessing

Alternative 40: 45.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (fma
    (- 3.0 (sqrt 5.0))
    (* 0.5 (cos y))
    (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * fma((3.0 - sqrt(5.0)), (0.5 * cos(y)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * fma(Float64(3.0 - sqrt(5.0)), Float64(0.5 * cos(y)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   4. flip-- N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   9. rem-square-sqrt N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   15. lower-+.f64 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   11. lower-sqrt.f64 57.6

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

7. Applied rewrites 57.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
11. Step-by-step derivation

    1. lift-+.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)}} \]

    2. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{2}{\sqrt{5} + 3} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \left(\color{blue}{\frac{2}{\sqrt{5} + 3} \cdot \cos y} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

    4. *-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \left(\color{blue}{\cos y \cdot \frac{2}{\sqrt{5} + 3}} + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)} \]

12. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
13. Add Preprocessing

Alternative 41: 45.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right) \cdot 0.5\right) \cdot 3 + 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (+
   (*
    (* (fma (- 3.0 (sqrt 5.0)) (cos y) (* (- (sqrt 5.0) 1.0) (cos x))) 0.5)
    3.0)
   3.0)))

C

double code(double x, double y) {
	return 2.0 / (((fma((3.0 - sqrt(5.0)), cos(y), ((sqrt(5.0) - 1.0) * cos(x))) * 0.5) * 3.0) + 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(Float64(Float64(fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))) * 0.5) * 3.0) + 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   4. flip-- N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   9. rem-square-sqrt N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   15. lower-+.f64 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   11. lower-sqrt.f64 57.6

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

7. Applied rewrites 57.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

12. Applied rewrites 44.3%

    \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right) \cdot 0.5\right) \cdot 3 + 3}} \]
13. Add Preprocessing

Alternative 42: 45.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (fma
    (fma (cos x) (- (sqrt 5.0) 1.0) (* (- 3.0 (sqrt 5.0)) (cos y)))
    0.5
    1.0))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * fma(fma(cos(x), (sqrt(5.0) - 1.0), ((3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0));
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * fma(fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 0.5, 1.0)))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right)}, \frac{1}{2}, 1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), \frac{1}{2}, 1\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}\right), \frac{1}{2}, 1\right)} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   13. lower-cos.f64 99.2

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \color{blue}{\cos y}\right), 0.5, 1\right)} \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]

   16. lower-cos.f64 61.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

8. Applied rewrites 61.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), \frac{1}{2}, 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 0.5, 1\right)} \]
12. Add Preprocessing

Alternative 43: 45.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right) \cdot 0.5, 3, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (* (fma (- 3.0 (sqrt 5.0)) (cos y) (* (- (sqrt 5.0) 1.0) (cos x))) 0.5)
   3.0
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma((fma((3.0 - sqrt(5.0)), cos(y), ((sqrt(5.0) - 1.0) * cos(x))) * 0.5), 3.0, 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(Float64(fma(Float64(3.0 - sqrt(5.0)), cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))) * 0.5), 3.0, 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   4. flip-- N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   9. rem-square-sqrt N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   15. lower-+.f64 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   11. lower-sqrt.f64 57.6

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

7. Applied rewrites 57.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

12. Applied rewrites 44.3%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right) \cdot 0.5, 3, 3\right)}} \]
13. Add Preprocessing

Alternative 44: 43.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{\sqrt{5} + 3} + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (fma
    (* 0.5 (cos x))
    (- (sqrt 5.0) 1.0)
    (+ (/ 2.0 (+ (sqrt 5.0) 3.0)) 1.0)))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), ((2.0 / (sqrt(5.0) + 3.0)) + 1.0)));
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), Float64(Float64(2.0 / Float64(sqrt(5.0) + 3.0)) + 1.0))))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   4. flip-- N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   9. rem-square-sqrt N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   15. lower-+.f64 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   11. lower-sqrt.f64 57.6

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

7. Applied rewrites 57.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

11. Taylor expanded in y around 0

    \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

    2. associate-+l+ N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \left(2 \cdot \frac{1}{3 + \sqrt{5}} + 1\right)\right)}} \]

    3. associate-*r* N/A

       \[\leadsto \frac{2}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \cos x\right) \cdot \left(\sqrt{5} - 1\right)} + \left(2 \cdot \frac{1}{3 + \sqrt{5}} + 1\right)\right)} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, 2 \cdot \frac{1}{3 + \sqrt{5}} + 1\right)}} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \cos x}, \sqrt{5} - 1, 2 \cdot \frac{1}{3 + \sqrt{5}} + 1\right)} \]

    6. lower-cos.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \color{blue}{\cos x}, \sqrt{5} - 1, 2 \cdot \frac{1}{3 + \sqrt{5}} + 1\right)} \]

    7. lower--.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \color{blue}{\sqrt{5} - 1}, 2 \cdot \frac{1}{3 + \sqrt{5}} + 1\right)} \]

    8. lower-sqrt.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \color{blue}{\sqrt{5}} - 1, 2 \cdot \frac{1}{3 + \sqrt{5}} + 1\right)} \]

    9. lower-+.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, \color{blue}{2 \cdot \frac{1}{3 + \sqrt{5}} + 1}\right)} \]

    10. associate-*r/ N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, \color{blue}{\frac{2 \cdot 1}{3 + \sqrt{5}}} + 1\right)} \]

    11. metadata-eval N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, \frac{\color{blue}{2}}{3 + \sqrt{5}} + 1\right)} \]

    12. lower-/.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, \color{blue}{\frac{2}{3 + \sqrt{5}}} + 1\right)} \]

    13. +-commutative N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, \frac{2}{\color{blue}{\sqrt{5} + 3}} + 1\right)} \]

    14. lower-+.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2} \cdot \cos x, \sqrt{5} - 1, \frac{2}{\color{blue}{\sqrt{5} + 3}} + 1\right)} \]

    15. lower-sqrt.f64 42.0

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{\color{blue}{\sqrt{5}} + 3} + 1\right)} \]

13. Applied rewrites 42.0%

    \[\leadsto \frac{2}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{\sqrt{5} + 3} + 1\right)}} \]
14. Add Preprocessing

Alternative 45: 43.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (fma (* 0.5 (cos x)) (- (sqrt 5.0) 1.0) (/ 2.0 (+ (sqrt 5.0) 3.0)))
   3.0
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(fma((0.5 * cos(x)), (sqrt(5.0) - 1.0), (2.0 / (sqrt(5.0) + 3.0))), 3.0, 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(fma(Float64(0.5 * cos(x)), Float64(sqrt(5.0) - 1.0), Float64(2.0 / Float64(sqrt(5.0) + 3.0))), 3.0, 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   4. flip-- N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   9. rem-square-sqrt N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   15. lower-+.f64 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   11. lower-sqrt.f64 57.6

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

7. Applied rewrites 57.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

11. Taylor expanded in y around 0

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)\right)}} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) + 1\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 2 \cdot \frac{1}{3 + \sqrt{5}}, 3, 3\right)}} \]

13. Applied rewrites 42.0%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \cos x, \sqrt{5} - 1, \frac{2}{\sqrt{5} + 3}\right), 3, 3\right)}} \]
14. Add Preprocessing

Alternative 46: 42.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right), 3, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (fma (/ (cos y) (+ (sqrt 5.0) 3.0)) 2.0 (fma 0.5 (sqrt 5.0) -0.5))
   3.0
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(fma((cos(y) / (sqrt(5.0) + 3.0)), 2.0, fma(0.5, sqrt(5.0), -0.5)), 3.0, 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(fma(Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), 2.0, fma(0.5, sqrt(5.0), -0.5)), 3.0, 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   4. flip-- N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right) \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   9. rem-square-sqrt N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

   15. lower-+.f64 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

   11. lower-sqrt.f64 57.6

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

7. Applied rewrites 57.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{2}{\sqrt{5} + 3} \cdot \cos y\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) + 1\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) \cdot 3 + 1 \cdot 3}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right) \cdot 3 + \color{blue}{3}} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}, 3, 3\right)}} \]

13. Applied rewrites 41.4%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 2, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right), 3, 3\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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))))))