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

Percentage Accurate: 99.3% → 99.2%

Time: 11.7s

Alternatives: 33

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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_1 := 1 - \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\\ \frac{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right) + 2}{\frac{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 3, t\_1, \left(\left(1 - \left(\left(t\_0 \cdot \cos x\right) \cdot \cos x\right) \cdot t\_0\right) \cdot 3\right) \cdot 2\right)}{t\_1 \cdot 2}} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (sqrt 5.0) -0.5))
        (t_1 (- 1.0 (* (fma (sqrt 5.0) 0.5 -0.5) (cos x)))))
   (/
    (+
     (*
      (- (cos x) (cos y))
      (*
       (- (sin y) (/ (sin x) 16.0))
       (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))))
     2.0)
    (/
     (fma
      (* (* (- 3.0 (sqrt 5.0)) (cos y)) 3.0)
      t_1
      (* (* (- 1.0 (* (* (* t_0 (cos x)) (cos x)) t_0)) 3.0) 2.0))
     (* t_1 2.0)))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, sqrt(5.0), -0.5);
	double t_1 = 1.0 - (fma(sqrt(5.0), 0.5, -0.5) * cos(x));
	return (((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)))) + 2.0) / (fma((((3.0 - sqrt(5.0)) * cos(y)) * 3.0), t_1, (((1.0 - (((t_0 * cos(x)) * cos(x)) * t_0)) * 3.0) * 2.0)) / (t_1 * 2.0));
}

Julia

function code(x, y)
	t_0 = fma(0.5, sqrt(5.0), -0.5)
	t_1 = Float64(1.0 - Float64(fma(sqrt(5.0), 0.5, -0.5) * cos(x)))
	return Float64(Float64(Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)))) + 2.0) / Float64(fma(Float64(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)) * 3.0), t_1, Float64(Float64(Float64(1.0 - Float64(Float64(Float64(t_0 * cos(x)) * cos(x)) * t_0)) * 3.0) * 2.0)) / Float64(t_1 * 2.0)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

   14. lift-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ t_1 := 1 - \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{\frac{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 3, t\_1, \left(\left(1 - \left(\left(t\_0 \cdot \cos x\right) \cdot \cos x\right) \cdot t\_0\right) \cdot 3\right) \cdot 2\right)}{t\_1 \cdot 2}} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (sqrt 5.0) -0.5))
        (t_1 (- 1.0 (* (fma (sqrt 5.0) 0.5 -0.5) (cos x)))))
   (/
    (fma
     (fma (sin y) -0.0625 (sin x))
     (* (* (fma (sin x) -0.0625 (sin y)) (- (cos x) (cos y))) (sqrt 2.0))
     2.0)
    (/
     (fma
      (* (* (- 3.0 (sqrt 5.0)) (cos y)) 3.0)
      t_1
      (* (* (- 1.0 (* (* (* t_0 (cos x)) (cos x)) t_0)) 3.0) 2.0))
     (* t_1 2.0)))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, sqrt(5.0), -0.5);
	double t_1 = 1.0 - (fma(sqrt(5.0), 0.5, -0.5) * cos(x));
	return fma(fma(sin(y), -0.0625, sin(x)), ((fma(sin(x), -0.0625, sin(y)) * (cos(x) - cos(y))) * sqrt(2.0)), 2.0) / (fma((((3.0 - sqrt(5.0)) * cos(y)) * 3.0), t_1, (((1.0 - (((t_0 * cos(x)) * cos(x)) * t_0)) * 3.0) * 2.0)) / (t_1 * 2.0));
}

Julia

function code(x, y)
	t_0 = fma(0.5, sqrt(5.0), -0.5)
	t_1 = Float64(1.0 - Float64(fma(sqrt(5.0), 0.5, -0.5) * cos(x)))
	return Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(fma(sin(x), -0.0625, sin(y)) * Float64(cos(x) - cos(y))) * sqrt(2.0)), 2.0) / Float64(fma(Float64(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)) * 3.0), t_1, Float64(Float64(Float64(1.0 - Float64(Float64(Float64(t_0 * cos(x)) * cos(x)) * t_0)) * 3.0) * 2.0)) / Float64(t_1 * 2.0)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

   14. lift-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

8. Applied rewrites 99.3%

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

9. Final simplification 99.3%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\\ t_1 := 1 - t\_0\\ \frac{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right) + 2}{\frac{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot 3, t\_1, \left(\left(1 - {t\_0}^{2}\right) \cdot 3\right) \cdot 2\right)}{t\_1 \cdot 2}} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (sqrt 5.0) 0.5 -0.5) (cos x))) (t_1 (- 1.0 t_0)))
   (/
    (+
     (*
      (- (cos x) (cos y))
      (*
       (- (sin y) (/ (sin x) 16.0))
       (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))))
     2.0)
    (/
     (fma
      (* (* (- 3.0 (sqrt 5.0)) (cos y)) 3.0)
      t_1
      (* (* (- 1.0 (pow t_0 2.0)) 3.0) 2.0))
     (* t_1 2.0)))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5) * cos(x);
	double t_1 = 1.0 - t_0;
	return (((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)))) + 2.0) / (fma((((3.0 - sqrt(5.0)) * cos(y)) * 3.0), t_1, (((1.0 - pow(t_0, 2.0)) * 3.0) * 2.0)) / (t_1 * 2.0));
}

Julia

function code(x, y)
	t_0 = Float64(fma(sqrt(5.0), 0.5, -0.5) * cos(x))
	t_1 = Float64(1.0 - t_0)
	return Float64(Float64(Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)))) + 2.0) / Float64(fma(Float64(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)) * 3.0), t_1, Float64(Float64(Float64(1.0 - (t_0 ^ 2.0)) * 3.0) * 2.0)) / Float64(t_1 * 2.0)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 99.2%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-+l+ N/A

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

6. Applied rewrites 99.2%

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

8. Applied rewrites 99.3%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 99.2%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. div-sub N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lift-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

6. Applied rewrites 99.3%

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

7. Final simplification 99.3%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-cos.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 8: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

4. Applied rewrites 99.2%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-+l+ N/A

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 9: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot t\_1\right) \cdot t\_0 + 2}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{2}{1 + \sqrt{5}} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.025:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0 \cdot \left(t\_1 \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2
         (/
          (+ (* (* (* (sin x) (sqrt 2.0)) t_1) t_0) 2.0)
          (*
           (+
            (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))
            (+ (* (/ 2.0 (+ 1.0 (sqrt 5.0))) (cos x)) 1.0))
           3.0))))
   (if (<= x -0.025)
     t_2
     (if (<= x 2e-5)
       (/
        (+ (* t_0 (* t_1 (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))) 2.0)
        (*
         (+
          (fma
           (- (sqrt 5.0) 1.0)
           (fma (* -0.25 x) x 0.5)
           (* (fma -0.5 (sqrt 5.0) 1.5) (cos y)))
          1.0)
         3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = ((((sin(x) * sqrt(2.0)) * t_1) * t_0) + 2.0) / (((((3.0 - sqrt(5.0)) / 2.0) * cos(y)) + (((2.0 / (1.0 + sqrt(5.0))) * cos(x)) + 1.0)) * 3.0);
	double tmp;
	if (x <= -0.025) {
		tmp = t_2;
	} else if (x <= 2e-5) {
		tmp = ((t_0 * (t_1 * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)))) + 2.0) / ((fma((sqrt(5.0) - 1.0), fma((-0.25 * x), x, 0.5), (fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(Float64(Float64(Float64(Float64(sin(x) * sqrt(2.0)) * t_1) * t_0) + 2.0) / Float64(Float64(Float64(Float64(Float64(3.0 - sqrt(5.0)) / 2.0) * cos(y)) + Float64(Float64(Float64(2.0 / Float64(1.0 + sqrt(5.0))) * cos(x)) + 1.0)) * 3.0))
	tmp = 0.0
	if (x <= -0.025)
		tmp = t_2;
	elseif (x <= 2e-5)
		tmp = Float64(Float64(Float64(t_0 * Float64(t_1 * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)))) + 2.0) / Float64(Float64(fma(Float64(sqrt(5.0) - 1.0), fma(Float64(-0.25 * x), x, 0.5), Float64(fma(-0.5, sqrt(5.0), 1.5) * cos(y))) + 1.0) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.025], t$95$2, If[LessEqual[x, 2e-5], N[(N[(N[(t$95$0 * N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[(-0.25 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] + N[(N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.025000000000000001 or 2.00000000000000016e-5 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.1

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

   5. Applied rewrites 64.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if -0.025000000000000001 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.025:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{2}{1 + \sqrt{5}} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right) + 2}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25 \cdot x, x, 0.5\right), \mathsf{fma}\left(-0.5, \sqrt{5}, 1.5\right) \cdot \cos y\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{2}{1 + \sqrt{5}} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\left(\frac{t\_1}{2} \cdot \cos y + \left(\frac{2}{1 + \sqrt{5}} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.18:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.052:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot t\_1 + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (+
           (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0)
           2.0)
          (*
           (+
            (* (/ t_1 2.0) (cos y))
            (+ (* (/ 2.0 (+ 1.0 (sqrt 5.0))) (cos x)) 1.0))
           3.0))))
   (if (<= x -0.18)
     t_2
     (if (<= x 0.052)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 x (sin y)) (sqrt 2.0))
           (fma (sin y) -0.0625 (sin x)))
          t_0)
         2.0)
        (*
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (+ (* (* 0.5 (cos y)) t_1) 1.0))
         3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = ((((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / ((((t_1 / 2.0) * cos(y)) + (((2.0 / (1.0 + sqrt(5.0))) * cos(x)) + 1.0)) * 3.0);
	double tmp;
	if (x <= -0.18) {
		tmp = t_2;
	} else if (x <= 0.052) {
		tmp = ((((fma(-0.0625, x, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (((0.5 * cos(y)) * t_1) + 1.0)) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / Float64(Float64(Float64(Float64(t_1 / 2.0) * cos(y)) + Float64(Float64(Float64(2.0 / Float64(1.0 + sqrt(5.0))) * cos(x)) + 1.0)) * 3.0))
	tmp = 0.0
	if (x <= -0.18)
		tmp = t_2;
	elseif (x <= 0.052)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, x, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(Float64(Float64(0.5 * cos(y)) * t_1) + 1.0)) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(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] + 2.0), $MachinePrecision] / N[(N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 / N[(1.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.18], t$95$2, If[LessEqual[x, 0.052], N[(N[(N[(N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17999999999999999 or 0.0519999999999999976 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.5

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

   5. Applied rewrites 63.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 63.6

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

   7. Applied rewrites 63.6%

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

   if -0.17999999999999999 < x < 0.0519999999999999976

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-+l+ N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.18:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{2}{1 + \sqrt{5}} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.052:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(\frac{2}{1 + \sqrt{5}} \cdot \cos x + 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := \sin x \cdot \sqrt{2}\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.18:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, t\_2 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\left(\left(\frac{t\_1}{2} \cdot \cos x + 1\right) + \frac{t\_3}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot t\_3 + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_2 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos y, t\_1 \cdot \cos x\right), 1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (* (sin x) (sqrt 2.0)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.18)
     (/
      (fma t_0 (* t_2 (fma -0.0625 (sin x) (sin y))) 2.0)
      (* (+ (+ (* (/ t_1 2.0) (cos x)) 1.0) (* (/ t_3 2.0) (cos y))) 3.0))
     (if (<= x 2e-5)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 x (sin y)) (sqrt 2.0))
           (fma (sin y) -0.0625 (sin x)))
          t_0)
         2.0)
        (*
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (+ (* (* 0.5 (cos y)) t_3) 1.0))
         3.0))
       (/
        (+ (* (* t_2 (- (sin y) (/ (sin x) 16.0))) t_0) 2.0)
        (* (fma 0.5 (fma t_3 (cos y) (* t_1 (cos x))) 1.0) 3.0))))))

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 = sin(x) * sqrt(2.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.18) {
		tmp = fma(t_0, (t_2 * fma(-0.0625, sin(x), sin(y))), 2.0) / (((((t_1 / 2.0) * cos(x)) + 1.0) + ((t_3 / 2.0) * cos(y))) * 3.0);
	} else if (x <= 2e-5) {
		tmp = ((((fma(-0.0625, x, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (((0.5 * cos(y)) * t_3) + 1.0)) * 3.0);
	} else {
		tmp = (((t_2 * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / (fma(0.5, fma(t_3, cos(y), (t_1 * cos(x))), 1.0) * 3.0);
	}
	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(sin(x) * sqrt(2.0))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.18)
		tmp = Float64(fma(t_0, Float64(t_2 * fma(-0.0625, sin(x), sin(y))), 2.0) / Float64(Float64(Float64(Float64(Float64(t_1 / 2.0) * cos(x)) + 1.0) + Float64(Float64(t_3 / 2.0) * cos(y))) * 3.0));
	elseif (x <= 2e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, x, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(Float64(Float64(0.5 * cos(y)) * t_3) + 1.0)) * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(t_2 * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / Float64(fma(0.5, fma(t_3, cos(y), Float64(t_1 * cos(x))), 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.18], N[(N[(t$95$0 * N[(t$95$2 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$3 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(N[(N[(N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$3 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17999999999999999

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 66.1

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

   5. Applied rewrites 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1

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

   7. Applied rewrites 66.1%

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

   if -0.17999999999999999 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-+l+ N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 2.00000000000000016e-5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-cos.f64 61.9

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

   8. Applied rewrites 61.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.18:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.18:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot t\_1 + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (+
           (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0)
           2.0)
          (*
           (fma 0.5 (fma t_1 (cos y) (* (- (sqrt 5.0) 1.0) (cos x))) 1.0)
           3.0))))
   (if (<= x -0.18)
     t_2
     (if (<= x 2e-5)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 x (sin y)) (sqrt 2.0))
           (fma (sin y) -0.0625 (sin x)))
          t_0)
         2.0)
        (*
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (+ (* (* 0.5 (cos y)) t_1) 1.0))
         3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = ((((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / (fma(0.5, fma(t_1, cos(y), ((sqrt(5.0) - 1.0) * cos(x))), 1.0) * 3.0);
	double tmp;
	if (x <= -0.18) {
		tmp = t_2;
	} else if (x <= 2e-5) {
		tmp = ((((fma(-0.0625, x, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (((0.5 * cos(y)) * t_1) + 1.0)) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / Float64(fma(0.5, fma(t_1, cos(y), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 1.0) * 3.0))
	tmp = 0.0
	if (x <= -0.18)
		tmp = t_2;
	elseif (x <= 2e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, x, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(Float64(Float64(0.5 * cos(y)) * t_1) + 1.0)) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(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] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.18], t$95$2, If[LessEqual[x, 2e-5], N[(N[(N[(N[(N[(N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17999999999999999 or 2.00000000000000016e-5 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-cos.f64 64.1

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

   8. Applied rewrites 64.1%

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

   if -0.17999999999999999 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-+l+ N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.18:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, x, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := \frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_2 \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_2\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3
         (/
          (+
           (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0)
           2.0)
          (* (fma 0.5 (fma t_1 (cos y) (* t_2 (cos x))) 1.0) 3.0))))
   (if (<= x -0.000145)
     t_3
     (if (<= x 1.4e-6)
       (/
        (+
         (*
          (*
           (* (fma (sin x) -0.0625 (sin y)) (sqrt 2.0))
           (fma (sin y) -0.0625 (sin x)))
          t_0)
         2.0)
        (* (fma 0.5 (fma t_1 (cos y) t_2) 1.0) 3.0))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double t_3 = ((((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / (fma(0.5, fma(t_1, cos(y), (t_2 * cos(x))), 1.0) * 3.0);
	double tmp;
	if (x <= -0.000145) {
		tmp = t_3;
	} else if (x <= 1.4e-6) {
		tmp = ((((fma(sin(x), -0.0625, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / (fma(0.5, fma(t_1, cos(y), t_2), 1.0) * 3.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	t_3 = Float64(Float64(Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / Float64(fma(0.5, fma(t_1, cos(y), Float64(t_2 * cos(x))), 1.0) * 3.0))
	tmp = 0.0
	if (x <= -0.000145)
		tmp = t_3;
	elseif (x <= 1.4e-6)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(sin(x), -0.0625, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / Float64(fma(0.5, fma(t_1, cos(y), t_2), 1.0) * 3.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(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] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.000145], t$95$3, If[LessEqual[x, 1.4e-6], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-4 or 1.39999999999999994e-6 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-cos.f64 64.1

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

   8. Applied rewrites 64.1%

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

   if -1.45e-4 < x < 1.39999999999999994e-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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \frac{\left(\left(\sin y \cdot \sqrt{2}\right) \cdot t\_1\right) \cdot t\_0 + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot t\_2 + 1\right) \cdot 3}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sqrt{2}\right) \cdot t\_1\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_2\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (fma (sin y) -0.0625 (sin x)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3
         (/
          (+ (* (* (* (sin y) (sqrt 2.0)) t_1) t_0) 2.0)
          (*
           (fma
            (fma 0.5 (sqrt 5.0) -0.5)
            (cos x)
            (+ (* (* 0.5 (cos y)) t_2) 1.0))
           3.0))))
   (if (<= y -1.32e-5)
     t_3
     (if (<= y 1.06e-19)
       (/
        (+ (* (* (* (fma (sin x) -0.0625 (sin y)) (sqrt 2.0)) t_1) t_0) 2.0)
        (* (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_2) 1.0) 3.0))
       t_3))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma(sin(y), -0.0625, sin(x));
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = ((((sin(y) * sqrt(2.0)) * t_1) * t_0) + 2.0) / (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (((0.5 * cos(y)) * t_2) + 1.0)) * 3.0);
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_3;
	} else if (y <= 1.06e-19) {
		tmp = ((((fma(sin(x), -0.0625, sin(y)) * sqrt(2.0)) * t_1) * t_0) + 2.0) / (fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), t_2), 1.0) * 3.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(sin(y), -0.0625, sin(x))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(Float64(Float64(Float64(sin(y) * sqrt(2.0)) * t_1) * t_0) + 2.0) / Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(Float64(Float64(0.5 * cos(y)) * t_2) + 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = t_3;
	elseif (y <= 1.06e-19)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(sin(x), -0.0625, sin(y)) * sqrt(2.0)) * t_1) * t_0) + 2.0) / Float64(fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_2), 1.0) * 3.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], t$95$3, If[LessEqual[y, 1.06e-19], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000007e-5 or 1.06e-19 < 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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-+l+ N/A

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if -1.32000000000000007e-5 < y < 1.06e-19

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\left(\sin y \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \left(\left(\frac{t\_1}{2} \cdot \cos x + 1\right) + \frac{t\_2}{2} \cdot \cos y\right) \cdot 3\\ \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_0 + 2}{t\_3}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos y, t\_1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x - 1\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{t\_3}\\ \end{array} \end{array} \]

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)))
        (t_3
         (* (+ (+ (* (/ t_1 2.0) (cos x)) 1.0) (* (/ t_2 2.0) (cos y))) 3.0)))
   (if (<= x -0.000145)
     (/ (+ (* (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)) t_0) 2.0) t_3)
     (if (<= x 1.4e-6)
       (/
        (+
         (*
          (*
           (* (fma (sin x) -0.0625 (sin y)) (sqrt 2.0))
           (fma (sin y) -0.0625 (sin x)))
          t_0)
         2.0)
        (* (fma 0.5 (fma t_2 (cos y) t_1) 1.0) 3.0))
       (/
        (+
         (*
          (- (cos x) 1.0)
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))
         2.0)
        t_3)))))

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 t_3 = ((((t_1 / 2.0) * cos(x)) + 1.0) + ((t_2 / 2.0) * cos(y))) * 3.0;
	double tmp;
	if (x <= -0.000145) {
		tmp = ((((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / t_3;
	} else if (x <= 1.4e-6) {
		tmp = ((((fma(sin(x), -0.0625, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / (fma(0.5, fma(t_2, cos(y), t_1), 1.0) * 3.0);
	} else {
		tmp = (((cos(x) - 1.0) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0)))) + 2.0) / t_3;
	}
	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))
	t_3 = Float64(Float64(Float64(Float64(Float64(t_1 / 2.0) * cos(x)) + 1.0) + Float64(Float64(t_2 / 2.0) * cos(y))) * 3.0)
	tmp = 0.0
	if (x <= -0.000145)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / t_3);
	elseif (x <= 1.4e-6)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(sin(x), -0.0625, sin(y)) * sqrt(2.0)) * fma(sin(y), -0.0625, sin(x))) * t_0) + 2.0) / Float64(fma(0.5, fma(t_2, cos(y), t_1), 1.0) * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(cos(x) - 1.0) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))) + 2.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, If[LessEqual[x, -0.000145], N[(N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[x, 1.4e-6], N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$2 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * 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]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-4

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if -1.45e-4 < x < 1.39999999999999994e-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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 1.39999999999999994e-6 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 58.9

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

   8. Applied rewrites 58.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x - 1\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\left(\frac{t\_0}{2} \cdot \cos x + 1\right) + \frac{t\_1}{2} \cdot \cos y\right) \cdot 3\\ \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{t\_2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot t\_1\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x - 1\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (* (+ (+ (* (/ t_0 2.0) (cos x)) 1.0) (* (/ t_1 2.0) (cos y))) 3.0)))
   (if (<= x -0.00325)
     (/
      (+
       (* (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)) (- (cos x) (cos y)))
       2.0)
      t_2)
     (if (<= x 2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (*
          (fma (* 1.00390625 (sin y)) x (* (pow (sin y) 2.0) -0.0625))
          (sqrt 2.0))
         2.0)
        (*
         (+ (fma t_0 (fma -0.25 (* x x) 0.5) (* (* 0.5 (cos y)) t_1)) 1.0)
         3.0))
       (/
        (+
         (*
          (- (cos x) 1.0)
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))
         2.0)
        t_2)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = ((((t_0 / 2.0) * cos(x)) + 1.0) + ((t_1 / 2.0) * cos(y))) * 3.0;
	double tmp;
	if (x <= -0.00325) {
		tmp = ((((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / t_2;
	} else if (x <= 2e-5) {
		tmp = fma((1.0 - cos(y)), (fma((1.00390625 * sin(y)), x, (pow(sin(y), 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / ((fma(t_0, fma(-0.25, (x * x), 0.5), ((0.5 * cos(y)) * t_1)) + 1.0) * 3.0);
	} else {
		tmp = (((cos(x) - 1.0) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0)))) + 2.0) / t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(Float64(Float64(Float64(t_0 / 2.0) * cos(x)) + 1.0) + Float64(Float64(t_1 / 2.0) * cos(y))) * 3.0)
	tmp = 0.0
	if (x <= -0.00325)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / t_2);
	elseif (x <= 2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(fma(Float64(1.00390625 * sin(y)), x, Float64((sin(y) ^ 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / Float64(Float64(fma(t_0, fma(-0.25, Float64(x * x), 0.5), Float64(Float64(0.5 * cos(y)) * t_1)) + 1.0) * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(cos(x) - 1.0) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))) + 2.0) / t_2);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]}, If[LessEqual[x, -0.00325], N[(N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * 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]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00324999999999999985

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if -0.00324999999999999985 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 58.9

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

   8. Applied rewrites 58.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x - 1\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_0 + 2}{\left(\left(\frac{t\_1}{2} \cdot \cos x + 1\right) + \frac{t\_2}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot t\_2\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1\right) \cdot 3}\\ \end{array} \end{array} \]

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 (<= x -0.00325)
     (/
      (+ (* (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)) t_0) 2.0)
      (* (+ (+ (* (/ t_1 2.0) (cos x)) 1.0) (* (/ t_2 2.0) (cos y))) 3.0))
     (if (<= x 2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (*
          (fma (* 1.00390625 (sin y)) x (* (pow (sin y) 2.0) -0.0625))
          (sqrt 2.0))
         2.0)
        (*
         (+ (fma t_1 (fma -0.25 (* x x) 0.5) (* (* 0.5 (cos y)) t_2)) 1.0)
         3.0))
       (/
        (+ (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0) 2.0)
        (* (fma 0.5 (fma t_1 (cos x) t_2) 1.0) 3.0))))))

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 (x <= -0.00325) {
		tmp = ((((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / (((((t_1 / 2.0) * cos(x)) + 1.0) + ((t_2 / 2.0) * cos(y))) * 3.0);
	} else if (x <= 2e-5) {
		tmp = fma((1.0 - cos(y)), (fma((1.00390625 * sin(y)), x, (pow(sin(y), 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / ((fma(t_1, fma(-0.25, (x * x), 0.5), ((0.5 * cos(y)) * t_2)) + 1.0) * 3.0);
	} else {
		tmp = ((((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / (fma(0.5, fma(t_1, cos(x), t_2), 1.0) * 3.0);
	}
	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 (x <= -0.00325)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / Float64(Float64(Float64(Float64(Float64(t_1 / 2.0) * cos(x)) + 1.0) + Float64(Float64(t_2 / 2.0) * cos(y))) * 3.0));
	elseif (x <= 2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(fma(Float64(1.00390625 * sin(y)), x, Float64((sin(y) ^ 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / Float64(Float64(fma(t_1, fma(-0.25, Float64(x * x), 0.5), Float64(Float64(0.5 * cos(y)) * t_2)) + 1.0) * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / Float64(fma(0.5, fma(t_1, cos(x), t_2), 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00325], N[(N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(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] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00324999999999999985

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sqrt.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if -0.00324999999999999985 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 58.7

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

   8. Applied rewrites 58.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \left(0.5 \cdot \cos y\right) \cdot t\_2\\ \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_0 + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, t\_3 + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), t\_3\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_2\right), 1\right) \cdot 3}\\ \end{array} \end{array} \]

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)))
        (t_3 (* (* 0.5 (cos y)) t_2)))
   (if (<= x -0.00325)
     (/
      (+ (* (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)) t_0) 2.0)
      (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (+ t_3 1.0)) 3.0))
     (if (<= x 2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (*
          (fma (* 1.00390625 (sin y)) x (* (pow (sin y) 2.0) -0.0625))
          (sqrt 2.0))
         2.0)
        (* (+ (fma t_1 (fma -0.25 (* x x) 0.5) t_3) 1.0) 3.0))
       (/
        (+ (* (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))) t_0) 2.0)
        (* (fma 0.5 (fma t_1 (cos x) t_2) 1.0) 3.0))))))

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 t_3 = (0.5 * cos(y)) * t_2;
	double tmp;
	if (x <= -0.00325) {
		tmp = ((((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (t_3 + 1.0)) * 3.0);
	} else if (x <= 2e-5) {
		tmp = fma((1.0 - cos(y)), (fma((1.00390625 * sin(y)), x, (pow(sin(y), 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / ((fma(t_1, fma(-0.25, (x * x), 0.5), t_3) + 1.0) * 3.0);
	} else {
		tmp = ((((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / (fma(0.5, fma(t_1, cos(x), t_2), 1.0) * 3.0);
	}
	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))
	t_3 = Float64(Float64(0.5 * cos(y)) * t_2)
	tmp = 0.0
	if (x <= -0.00325)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(t_3 + 1.0)) * 3.0));
	elseif (x <= 2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(fma(Float64(1.00390625 * sin(y)), x, Float64((sin(y) ^ 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / Float64(Float64(fma(t_1, fma(-0.25, Float64(x * x), 0.5), t_3) + 1.0) * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / Float64(fma(0.5, fma(t_1, cos(x), t_2), 1.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[x, -0.00325], N[(N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(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] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00324999999999999985

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

   4. Applied rewrites 98.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-+l+ N/A

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 63.3

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

   9. Applied rewrites 63.3%

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

   if -0.00324999999999999985 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-out N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 58.7

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

   8. Applied rewrites 58.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := \left(0.5 \cdot \cos y\right) \cdot t\_2\\ \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left(t\_1 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, t\_3 + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), t\_3\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{\left(\left(\frac{t\_0}{2} \cdot \cos x + 1\right) + \frac{t\_2}{2} \cdot \cos y\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* (* 0.5 (cos y)) t_2)))
   (if (<= x -0.00325)
     (/
      (+ (* (* (* t_1 -0.0625) (sqrt 2.0)) (- (cos x) (cos y))) 2.0)
      (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (+ t_3 1.0)) 3.0))
     (if (<= x 2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (*
          (fma (* 1.00390625 (sin y)) x (* (pow (sin y) 2.0) -0.0625))
          (sqrt 2.0))
         2.0)
        (* (+ (fma t_0 (fma -0.25 (* x x) 0.5) t_3) 1.0) 3.0))
       (/
        (fma (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0)) t_1 2.0)
        (*
         (+ (+ (* (/ t_0 2.0) (cos x)) 1.0) (* (/ t_2 2.0) (cos y)))
         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 = 3.0 - sqrt(5.0);
	double t_3 = (0.5 * cos(y)) * t_2;
	double tmp;
	if (x <= -0.00325) {
		tmp = ((((t_1 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (t_3 + 1.0)) * 3.0);
	} else if (x <= 2e-5) {
		tmp = fma((1.0 - cos(y)), (fma((1.00390625 * sin(y)), x, (pow(sin(y), 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / ((fma(t_0, fma(-0.25, (x * x), 0.5), t_3) + 1.0) * 3.0);
	} else {
		tmp = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), t_1, 2.0) / (((((t_0 / 2.0) * cos(x)) + 1.0) + ((t_2 / 2.0) * cos(y))) * 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(3.0 - sqrt(5.0))
	t_3 = Float64(Float64(0.5 * cos(y)) * t_2)
	tmp = 0.0
	if (x <= -0.00325)
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(t_3 + 1.0)) * 3.0));
	elseif (x <= 2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(fma(Float64(1.00390625 * sin(y)), x, Float64((sin(y) ^ 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / Float64(Float64(fma(t_0, fma(-0.25, Float64(x * x), 0.5), t_3) + 1.0) * 3.0));
	else
		tmp = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), t_1, 2.0) / Float64(Float64(Float64(Float64(Float64(t_0 / 2.0) * cos(x)) + 1.0) + Float64(Float64(t_2 / 2.0) * cos(y))) * 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[x, -0.00325], N[(N[(N[(N[(N[(t$95$1 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$2 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00324999999999999985

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-/.f64 N/A

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

      10. div-inv N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{1}{-16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\mathsf{neg}\left(16\right)}, \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{1}{\color{blue}{-16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \color{blue}{\frac{-1}{16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   6. Applied rewrites 98.8%

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin x}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin x}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 63.3

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   9. Applied rewrites 63.3%

      \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -0.00324999999999999985 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left({\sin y}^{2} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)\right)} + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right)} + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right) + x \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   if 2.00000000000000016e-5 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 80.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\left(\left(\frac{t\_1}{2} \cdot \cos x + 1\right) + \frac{t\_0}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot t\_0\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (*
           (+ (+ (* (/ t_1 2.0) (cos x)) 1.0) (* (/ t_0 2.0) (cos y)))
           3.0))))
   (if (<= x -0.00325)
     t_2
     (if (<= x 2e-5)
       (/
        (fma
         (- 1.0 (cos y))
         (*
          (fma (* 1.00390625 (sin y)) x (* (pow (sin y) 2.0) -0.0625))
          (sqrt 2.0))
         2.0)
        (*
         (+ (fma t_1 (fma -0.25 (* x x) 0.5) (* (* 0.5 (cos y)) t_0)) 1.0)
         3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / (((((t_1 / 2.0) * cos(x)) + 1.0) + ((t_0 / 2.0) * cos(y))) * 3.0);
	double tmp;
	if (x <= -0.00325) {
		tmp = t_2;
	} else if (x <= 2e-5) {
		tmp = fma((1.0 - cos(y)), (fma((1.00390625 * sin(y)), x, (pow(sin(y), 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / ((fma(t_1, fma(-0.25, (x * x), 0.5), ((0.5 * cos(y)) * t_0)) + 1.0) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / Float64(Float64(Float64(Float64(Float64(t_1 / 2.0) * cos(x)) + 1.0) + Float64(Float64(t_0 / 2.0) * cos(y))) * 3.0))
	tmp = 0.0
	if (x <= -0.00325)
		tmp = t_2;
	elseif (x <= 2e-5)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(fma(Float64(1.00390625 * sin(y)), x, Float64((sin(y) ^ 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / Float64(Float64(fma(t_1, fma(-0.25, Float64(x * x), 0.5), Float64(Float64(0.5 * cos(y)) * t_0)) + 1.0) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00325], t$95$2, If[LessEqual[x, 2e-5], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$1 * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00324999999999999985 or 2.00000000000000016e-5 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 60.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -0.00324999999999999985 < x < 2.00000000000000016e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left({\sin y}^{2} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)\right)} + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right)} + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right) + x \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + 1\right)}} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00325:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\left(\mathsf{fma}\left(\sqrt{5} - 1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right) + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\left(\left(\frac{t\_1}{2} \cdot \cos x + 1\right) + \frac{t\_0}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{if}\;x \leq -0.00015:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos y, t\_1\right), 0.5, 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (*
           (+ (+ (* (/ t_1 2.0) (cos x)) 1.0) (* (/ t_0 2.0) (cos y)))
           3.0))))
   (if (<= x -0.00015)
     t_2
     (if (<= x 1.4e-6)
       (/
        (fma
         (- 1.0 (cos y))
         (*
          (fma (* 1.00390625 (sin y)) x (* (pow (sin y) 2.0) -0.0625))
          (sqrt 2.0))
         2.0)
        (* (fma (fma t_0 (cos y) t_1) 0.5 1.0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / (((((t_1 / 2.0) * cos(x)) + 1.0) + ((t_0 / 2.0) * cos(y))) * 3.0);
	double tmp;
	if (x <= -0.00015) {
		tmp = t_2;
	} else if (x <= 1.4e-6) {
		tmp = fma((1.0 - cos(y)), (fma((1.00390625 * sin(y)), x, (pow(sin(y), 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / (fma(fma(t_0, cos(y), t_1), 0.5, 1.0) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / Float64(Float64(Float64(Float64(Float64(t_1 / 2.0) * cos(x)) + 1.0) + Float64(Float64(t_0 / 2.0) * cos(y))) * 3.0))
	tmp = 0.0
	if (x <= -0.00015)
		tmp = t_2;
	elseif (x <= 1.4e-6)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(fma(Float64(1.00390625 * sin(y)), x, Float64((sin(y) ^ 2.0) * -0.0625)) * sqrt(2.0)), 2.0) / Float64(fma(fma(t_0, cos(y), t_1), 0.5, 1.0) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$1 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00015], t$95$2, If[LessEqual[x, 1.4e-6], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Cos[y], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999987e-4 or 1.39999999999999994e-6 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right), {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 60.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -1.49999999999999987e-4 < x < 1.39999999999999994e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left({\sin y}^{2} \cdot \sqrt{2}\right) \cdot \left(1 - \cos y\right)\right)} + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right)} + x \cdot \left(\sqrt{2} \cdot \left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \left(1 - \cos y\right)\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right) + x \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(1 - \cos y\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) \cdot \left(1 - \cos y\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)}\right)} \]

      2. associate-+r- N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) - 1\right)}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \left(1 + \frac{1}{2} \cdot \left(\color{blue}{\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} - 1\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) + 1\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1\right) \cdot \frac{1}{2}} + 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(\frac{257}{256} \cdot \sin y, x, {\sin y}^{2} \cdot \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) - 1, \frac{1}{2}, 1\right)}} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \cos y, \sqrt{2} \cdot \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00015:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(1.00390625 \cdot \sin y, x, {\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\frac{4}{\sqrt{5} + 3} \cdot \cos y, 0.5, t\_1\right) \cdot 3}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_2\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(t\_2 \cdot \cos y, 0.5, t\_1\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))
          (pow (sin y) 2.0)
          2.0))
        (t_1 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -3.3e-6)
     (/ t_0 (* (fma (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)) 0.5 t_1) 3.0))
     (if (<= y 1.06e-19)
       (*
        (/
         (fma
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          (pow (sin x) 2.0)
          2.0)
         (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_2) 1.0))
        0.3333333333333333)
       (/ t_0 (* (fma (* t_2 (cos y)) 0.5 t_1) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0);
	double t_1 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -3.3e-6) {
		tmp = t_0 / (fma(((4.0 / (sqrt(5.0) + 3.0)) * cos(y)), 0.5, t_1) * 3.0);
	} else if (y <= 1.06e-19) {
		tmp = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), t_2), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_0 / (fma((t_2 * cos(y)), 0.5, t_1) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0)
	t_1 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -3.3e-6)
		tmp = Float64(t_0 / Float64(fma(Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y)), 0.5, t_1) * 3.0));
	elseif (y <= 1.06e-19)
		tmp = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_2), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(t_0 / Float64(fma(Float64(t_2 * cos(y)), 0.5, t_1) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.3e-6], N[(t$95$0 / N[(N[(N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e-19], N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(t$95$0 / N[(N[(N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000017e-6

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}}{3 + \sqrt{5}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      5. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{3 \cdot 3 - \color{blue}{5}}{3 + \sqrt{5}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{9} - 5}{3 + \sqrt{5}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{\color{blue}{4}}{3 + \sqrt{5}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{4}{3 + \sqrt{5}}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{4}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y, \frac{1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)\right)} \]

      10. lower-+.f64 61.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{4}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   9. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\color{blue}{\frac{4}{\sqrt{5} + 3}} \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \]

   if -3.30000000000000017e-6 < y < 1.06e-19

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x - \frac{\sin y}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x + \left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right) + \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\sin y}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin y \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{-1}{16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{1}{-16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\mathsf{neg}\left(16\right)}, \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{1}{\color{blue}{-16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \color{blue}{\frac{-1}{16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if 1.06e-19 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{4}{\sqrt{5} + 3} \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_2 \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_0 \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))
          (pow (sin y) 2.0)
          2.0))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= y -3.3e-6)
     (/ t_1 (* (fma 0.5 (fma t_0 (cos y) (* t_2 (cos x))) 1.0) 3.0))
     (if (<= y 1.06e-19)
       (*
        (/
         (fma
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          (pow (sin x) 2.0)
          2.0)
         (fma 0.5 (fma t_2 (cos x) t_0) 1.0))
        0.3333333333333333)
       (/
        t_1
        (*
         (fma (* t_0 (cos y)) 0.5 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (y <= -3.3e-6) {
		tmp = t_1 / (fma(0.5, fma(t_0, cos(y), (t_2 * cos(x))), 1.0) * 3.0);
	} else if (y <= 1.06e-19) {
		tmp = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(0.5, fma(t_2, cos(x), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_1 / (fma((t_0 * cos(y)), 0.5, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0)
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (y <= -3.3e-6)
		tmp = Float64(t_1 / Float64(fma(0.5, fma(t_0, cos(y), Float64(t_2 * cos(x))), 1.0) * 3.0));
	elseif (y <= 1.06e-19)
		tmp = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(0.5, fma(t_2, cos(x), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = Float64(t_1 / Float64(fma(Float64(t_0 * cos(y)), 0.5, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[y, -3.3e-6], N[(t$95$1 / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e-19], N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(t$95$1 / N[(N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.30000000000000017e-6

   1. Initial program 98.8%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right)}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right)} \cdot \cos x\right), 1\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\color{blue}{\sqrt{5}} - 1\right) \cdot \cos x\right), 1\right)} \]

      14. lower-cos.f64 61.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \color{blue}{\cos x}\right), 1\right)} \]

   8. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}} \]

   if -3.30000000000000017e-6 < y < 1.06e-19

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x - \frac{\sin y}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x + \left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right) + \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\sin y}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin y \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{-1}{16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{1}{-16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\mathsf{neg}\left(16\right)}, \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{1}{\color{blue}{-16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \color{blue}{\frac{-1}{16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if 1.06e-19 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 58.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\frac{\sqrt{5} - 1}{2}} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\frac{\color{blue}{\sqrt{5} - 1}}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      7. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      13. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\color{blue}{\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)\right)} \]

   7. Applied rewrites 58.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} - 1\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, t\_1 \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2
         (/
          (fma
           (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))
           (pow (sin y) 2.0)
           2.0)
          (* (fma 0.5 (fma t_0 (cos y) (* t_1 (cos x))) 1.0) 3.0))))
   (if (<= y -3.3e-6)
     t_2
     (if (<= y 1.06e-19)
       (*
        (/
         (fma
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          (pow (sin x) 2.0)
          2.0)
         (fma 0.5 (fma t_1 (cos x) t_0) 1.0))
        0.3333333333333333)
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (fma(0.5, fma(t_0, cos(y), (t_1 * cos(x))), 1.0) * 3.0);
	double tmp;
	if (y <= -3.3e-6) {
		tmp = t_2;
	} else if (y <= 1.06e-19) {
		tmp = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(0.5, fma(t_1, cos(x), t_0), 1.0)) * 0.3333333333333333;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(fma(0.5, fma(t_0, cos(y), Float64(t_1 * cos(x))), 1.0) * 3.0))
	tmp = 0.0
	if (y <= -3.3e-6)
		tmp = t_2;
	elseif (y <= 1.06e-19)
		tmp = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(0.5, fma(t_1, cos(x), t_0), 1.0)) * 0.3333333333333333);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[(t$95$1 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.3e-6], t$95$2, If[LessEqual[y, 1.06e-19], N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.30000000000000017e-6 or 1.06e-19 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 59.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right)} + 1\right)} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \cos x \cdot \left(\sqrt{5} - 1\right), 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right)}, 1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \cos x \cdot \left(\sqrt{5} - 1\right)\right), 1\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x}\right), 1\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\left(\sqrt{5} - 1\right)} \cdot \cos x\right), 1\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\color{blue}{\sqrt{5}} - 1\right) \cdot \cos x\right), 1\right)} \]

      14. lower-cos.f64 60.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \color{blue}{\cos x}\right), 1\right)} \]

   8. Applied rewrites 60.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}} \]

   if -3.30000000000000017e-6 < y < 1.06e-19

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x - \frac{\sin y}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x + \left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right) + \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\sin y}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin y \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{-1}{16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{1}{-16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\mathsf{neg}\left(16\right)}, \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{1}{\color{blue}{-16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \color{blue}{\frac{-1}{16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\sin y}^{2}, \mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos y, t\_0\right), 0.5, 1\right)}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
            (pow (sin x) 2.0)
            2.0)
           (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
          0.3333333333333333)))
   (if (<= x -9.6e-5)
     t_2
     (if (<= x 7e-7)
       (/
        (/
         (fma
          (pow (sin y) 2.0)
          (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))
          2.0)
         (fma (fma t_1 (cos y) t_0) 0.5 1.0))
        3.0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -9.6e-5) {
		tmp = t_2;
	} else if (x <= 7e-7) {
		tmp = (fma(pow(sin(y), 2.0), (fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), 2.0) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0)) / 3.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -9.6e-5)
		tmp = t_2;
	elseif (x <= 7e-7)
		tmp = Float64(Float64(fma((sin(y) ^ 2.0), Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), 2.0) / fma(fma(t_1, cos(y), t_0), 0.5, 1.0)) / 3.0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -9.6e-5], t$95$2, If[LessEqual[x, 7e-7], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6000000000000002e-5 or 6.99999999999999968e-7 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x - \frac{\sin y}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x + \left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right) + \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\sin y}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin y \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{-1}{16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{1}{-16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\mathsf{neg}\left(16\right)}, \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{1}{\color{blue}{-16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \color{blue}{\frac{-1}{16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   6. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -9.6000000000000002e-5 < x < 6.99999999999999968e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

      10. lower-sqrt.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right) \cdot 3}} \]

   10. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left({\sin y}^{2}, \mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 0.5, 1\right)}}{3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos y, t\_0\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (/
           (fma
            (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
            (pow (sin x) 2.0)
            2.0)
           (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
          0.3333333333333333)))
   (if (<= x -9.6e-5)
     t_2
     (if (<= x 7e-7)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (* (fma 0.5 (fma t_1 (cos y) t_0) 1.0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -9.6e-5) {
		tmp = t_2;
	} else if (x <= 7e-7) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (fma(0.5, fma(t_1, cos(y), t_0), 1.0) * 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(0.5, fma(t_0, cos(x), t_1), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -9.6e-5)
		tmp = t_2;
	elseif (x <= 7e-7)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(fma(0.5, fma(t_1, cos(y), t_0), 1.0) * 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -9.6e-5], t$95$2, If[LessEqual[x, 7e-7], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$1 * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6000000000000002e-5 or 6.99999999999999968e-7 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x - \frac{\sin y}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x + \left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right) + \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\sin y}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin y \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{-1}{16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{1}{-16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\mathsf{neg}\left(16\right)}, \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{1}{\color{blue}{-16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \color{blue}{\frac{-1}{16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   6. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -9.6000000000000002e-5 < x < 6.99999999999999968e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 1\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

      10. lower-sqrt.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos y, \sqrt{5}\right), 0.5\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (/
           (fma
            (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
            (pow (sin x) 2.0)
            2.0)
           (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 1.0))
          0.3333333333333333)))
   (if (<= x -9.6e-5)
     t_1
     (if (<= x 7e-7)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (* (fma 0.5 (fma t_0 (cos y) (sqrt 5.0)) 0.5) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), t_0), 1.0)) * 0.3333333333333333;
	double tmp;
	if (x <= -9.6e-5) {
		tmp = t_1;
	} else if (x <= 7e-7) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (fma(0.5, fma(t_0, cos(y), sqrt(5.0)), 0.5) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 1.0)) * 0.3333333333333333)
	tmp = 0.0
	if (x <= -9.6e-5)
		tmp = t_1;
	elseif (x <= 7e-7)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(fma(0.5, fma(t_0, cos(y), sqrt(5.0)), 0.5) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[x, -9.6e-5], t$95$1, If[LessEqual[x, 7e-7], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 * N[Cos[y], $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6000000000000002e-5 or 6.99999999999999968e-7 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x - \frac{\sin y}{16}\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin x + \left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\sin y}{16}\right)\right) + \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\frac{\sin y}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. div-inv N/A

          \[\leadsto \frac{2 + \left(\left(\left(\mathsf{neg}\left(\color{blue}{\sin y \cdot \frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{2 + \left(\left(\color{blue}{\sin y \cdot \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{16}}\right)\right) + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{-1}{16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \color{blue}{\frac{1}{-16}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}} + \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\mathsf{neg}\left(16\right)}, \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{1}{\color{blue}{-16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \color{blue}{\frac{-1}{16}}, \sin x\right) \cdot \left(\sqrt{2} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. div-inv N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   6. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   9. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -9.6000000000000002e-5 < x < 6.99999999999999968e-7

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{e^{\log \left({\left(\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)\right)}^{-1}\right) \cdot -1}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \frac{1}{2}\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}}, \frac{1}{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \sqrt{5}, \frac{1}{2}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right)}, \frac{1}{2}\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5}\right), \frac{1}{2}\right)} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5}\right), \frac{1}{2}\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5}\right), \frac{1}{2}\right)} \]

      10. lower-sqrt.f64 98.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}}\right), 0.5\right)} \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 59.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
  (* (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) (sqrt 5.0)) 0.5) 3.0)))

C

double code(double x, double y) {
	return fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), sqrt(5.0)), 0.5) * 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), sqrt(5.0)), 0.5) * 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 61.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

7. Applied rewrites 61.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{e^{\log \left({\left(\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)\right)}^{-1}\right) \cdot -1}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \frac{1}{2}\right)}} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + \frac{1}{2}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}}, \frac{1}{2}\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \sqrt{5}, \frac{1}{2}\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right)}, \frac{1}{2}\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5}\right), \frac{1}{2}\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5}\right), \frac{1}{2}\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5}\right), \frac{1}{2}\right)} \]

   10. lower-sqrt.f64 58.2

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}}\right), 0.5\right)} \]

10. Applied rewrites 58.2%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right)}} \]

11. Final simplification 58.2%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right) \cdot 3} \]
12. Add Preprocessing

Alternative 29: 40.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{2 \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
  (* 2.0 3.0)))

C

double code(double x, double y) {
	return fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (2.0 * 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(2.0 * 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(2.0 * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 61.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

   10. lower-sqrt.f64 58.3

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

8. Applied rewrites 58.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot 2} \]
10. Step-by-step derivation

11. Applied rewrites 39.8%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot 2} \]

12. Final simplification 39.8%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{2 \cdot 3} \]
13. Add Preprocessing

Alternative 30: 34.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{\mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(-0.25, y \cdot y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (pow y 4.0) -0.03125) (sqrt 2.0) 2.0)
  (*
   (fma
    (- 3.0 (sqrt 5.0))
    (fma -0.25 (* y y) 0.5)
    (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
   3.0)))

C

double code(double x, double y) {
	return fma((pow(y, 4.0) * -0.03125), sqrt(2.0), 2.0) / (fma((3.0 - sqrt(5.0)), fma(-0.25, (y * y), 0.5), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64((y ^ 4.0) * -0.03125), sqrt(2.0), 2.0) / Float64(fma(Float64(3.0 - sqrt(5.0)), fma(-0.25, Float64(y * y), 0.5), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[y, 4.0], $MachinePrecision] * -0.03125), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 61.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

   10. lower-sqrt.f64 58.3

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

8. Applied rewrites 58.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{2 + \color{blue}{\frac{-1}{32} \cdot \left({y}^{4} \cdot \sqrt{2}\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 29.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
13. Step-by-step derivation

    1. associate-+r+ N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt{5} - 1\right) \cdot \cos x\right)} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    3. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) \cdot \cos x} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    4. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \cos x + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) \cdot \cos x + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    6. distribute-lft-in N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)} \cdot \cos x + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    7. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right) \cdot \cos x + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \cdot \cos x + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    9. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)} \cdot \cos x + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    10. *-commutative N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\color{blue}{\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    11. +-commutative N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\left(1 + \frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}\right)} \]

    12. associate-+r+ N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)\right)}} \]

14. Applied rewrites 32.9%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(-0.25, y \cdot y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}} \]

15. Final simplification 32.9%

    \[\leadsto \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{\mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(-0.25, y \cdot y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3} \]
16. Add Preprocessing

Alternative 31: 33.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (pow y 4.0) -0.03125) (sqrt 2.0) 2.0)
  (* (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 1.0) 3.0)))

C

double code(double x, double y) {
	return fma((pow(y, 4.0) * -0.03125), sqrt(2.0), 2.0) / (fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 1.0) * 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64((y ^ 4.0) * -0.03125), sqrt(2.0), 2.0) / Float64(fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 1.0) * 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[y, 4.0], $MachinePrecision] * -0.03125), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 61.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

   10. lower-sqrt.f64 58.3

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

8. Applied rewrites 58.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{2 + \color{blue}{\frac{-1}{32} \cdot \left({y}^{4} \cdot \sqrt{2}\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 29.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

    2. distribute-lft-out N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 1\right)}} \]

    4. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{5} - 1\right) \cdot \cos x} + \left(3 - \sqrt{5}\right), 1\right)} \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)}, 1\right)} \]

    6. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5} - 1}, \cos x, 3 - \sqrt{5}\right), 1\right)} \]

    7. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\sqrt{5}} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \]

    8. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \color{blue}{\cos x}, 3 - \sqrt{5}\right), 1\right)} \]

    9. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \color{blue}{3 - \sqrt{5}}\right), 1\right)} \]

    10. lower-sqrt.f64 31.7

        \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \color{blue}{\sqrt{5}}\right), 1\right)} \]

14. Applied rewrites 31.7%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)}} \]

15. Final simplification 31.7%

    \[\leadsto \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right) \cdot 3} \]
16. Add Preprocessing

Alternative 32: 31.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot -0.25, 3 - \sqrt{5}, 2\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (pow y 4.0) -0.03125) (sqrt 2.0) 2.0)
  (* (fma (* (* y y) -0.25) (- 3.0 (sqrt 5.0)) 2.0) 3.0)))

C

double code(double x, double y) {
	return fma((pow(y, 4.0) * -0.03125), sqrt(2.0), 2.0) / (fma(((y * y) * -0.25), (3.0 - sqrt(5.0)), 2.0) * 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64((y ^ 4.0) * -0.03125), sqrt(2.0), 2.0) / Float64(fma(Float64(Float64(y * y) * -0.25), Float64(3.0 - sqrt(5.0)), 2.0) * 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[y, 4.0], $MachinePrecision] * -0.03125), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(y * y), $MachinePrecision] * -0.25), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 61.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

   10. lower-sqrt.f64 58.3

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

8. Applied rewrites 58.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{2 + \color{blue}{\frac{-1}{32} \cdot \left({y}^{4} \cdot \sqrt{2}\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 29.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \left(2 + \color{blue}{\frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]
13. Step-by-step derivation

14. Applied rewrites 30.5%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(-0.25 \cdot \left(y \cdot y\right), \color{blue}{3 - \sqrt{5}}, 2\right)} \]

15. Final simplification 30.5%

    \[\leadsto \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot -0.25, 3 - \sqrt{5}, 2\right) \cdot 3} \]
16. Add Preprocessing

Alternative 33: 30.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{2 \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (fma (* (pow y 4.0) -0.03125) (sqrt 2.0) 2.0) (* 2.0 3.0)))

C

double code(double x, double y) {
	return fma((pow(y, 4.0) * -0.03125), sqrt(2.0), 2.0) / (2.0 * 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64((y ^ 4.0) * -0.03125), sqrt(2.0), 2.0) / Float64(2.0 * 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[y, 4.0], $MachinePrecision] * -0.03125), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(2.0 * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 61.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \left(\sqrt{5} - 1\right), 1\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5} - 1\right), 1\right)} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5} - 1}\right), 1\right)} \]

   10. lower-sqrt.f64 58.3

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}} - 1\right), 1\right)} \]

8. Applied rewrites 58.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{2 + \color{blue}{\frac{-1}{32} \cdot \left({y}^{4} \cdot \sqrt{2}\right)}}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]
10. Step-by-step derivation

11. Applied rewrites 29.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5} - 1\right), 1\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{32} \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot 2} \]
13. Step-by-step derivation

14. Applied rewrites 29.3%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \sqrt{2}, 2\right)}{3 \cdot 2} \]

15. Final simplification 29.3%

    \[\leadsto \frac{\mathsf{fma}\left({y}^{4} \cdot -0.03125, \sqrt{2}, 2\right)}{2 \cdot 3} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))))))