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

Percentage Accurate: 99.3% → 99.3%

Time: 20.5s

Alternatives: 32

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right) + 2}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \end{array} \]

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
   (/ 6.0 (+ 3.0 (sqrt 5.0)))
   (cos y)
   (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 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((6.0 / (3.0 + sqrt(5.0))), cos(y), (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 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(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 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[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. rem-square-sqrt N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lift--.f64 N/A

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

   7. flip-- N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. rem-square-sqrt N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f64 N/A

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

   18. metadata-eval N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.5

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in y around inf

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

7. Applied rewrites 99.4%

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

Alternative 6: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_3 := \sin x \cdot \sqrt{2}\\ t_4 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -0.0025:\\ \;\;\;\;\frac{\left(t\_3 \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_1 + 2}{\left(\frac{2}{t\_4} \cdot \cos y + \left(\frac{t\_0}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot t\_2\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{t\_4}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_3 \cdot t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0 \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2 (fma (sin x) -0.0625 (sin y)))
        (t_3 (* (sin x) (sqrt 2.0)))
        (t_4 (+ 3.0 (sqrt 5.0))))
   (if (<= x -0.0025)
     (/
      (+ (* (* t_3 (- (sin y) (/ (sin x) 16.0))) t_1) 2.0)
      (* (+ (* (/ 2.0 t_4) (cos y)) (+ (* (/ t_0 2.0) (cos x)) 1.0)) 3.0))
     (if (<= x 0.0038)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (- 1.0 (cos y)) t_2) (sqrt 2.0))
         2.0)
        (fma
         (/ 6.0 t_4)
         (cos y)
         (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 3.0)))
       (/
        (fma t_1 (* t_3 t_2) 2.0)
        (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (* t_0 (cos x))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(sin(x), -0.0625, sin(y));
	double t_3 = sin(x) * sqrt(2.0);
	double t_4 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -0.0025) {
		tmp = (((t_3 * (sin(y) - (sin(x) / 16.0))) * t_1) + 2.0) / ((((2.0 / t_4) * cos(y)) + (((t_0 / 2.0) * cos(x)) + 1.0)) * 3.0);
	} else if (x <= 0.0038) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (((1.0 - cos(y)) * t_2) * sqrt(2.0)), 2.0) / fma((6.0 / t_4), cos(y), (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0));
	} else {
		tmp = fma(t_1, (t_3 * t_2), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (t_0 * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = fma(sin(x), -0.0625, sin(y))
	t_3 = Float64(sin(x) * sqrt(2.0))
	t_4 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0025)
		tmp = Float64(Float64(Float64(Float64(t_3 * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1) + 2.0) / Float64(Float64(Float64(Float64(2.0 / t_4) * cos(y)) + Float64(Float64(Float64(t_0 / 2.0) * cos(x)) + 1.0)) * 3.0));
	elseif (x <= 0.0038)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * t_2) * sqrt(2.0)), 2.0) / fma(Float64(6.0 / t_4), cos(y), Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0)));
	else
		tmp = Float64(fma(t_1, Float64(t_3 * t_2), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(t_0 * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0025], N[(N[(N[(N[(t$95$3 * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(2.0 / t$95$4), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 / 2.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0038], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / t$95$4), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$3 * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00250000000000000005

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-*r/ N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 66.0

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

   7. Applied rewrites 66.0%

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

   if -0.00250000000000000005 < x < 0.00379999999999999999

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 99.7

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

   11. Applied rewrites 99.7%

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

   if 0.00379999999999999999 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 74.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0025:\\ \;\;\;\;\frac{\left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\left(\frac{2}{3 + \sqrt{5}} \cdot \cos y + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_1 := \mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot t\_0, 2\right)\\ t_2 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -0.0025:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(2, \frac{\cos y}{t\_2}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot t\_0\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{t\_2}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin x) -0.0625 (sin y)))
        (t_1 (fma (- (cos x) (cos y)) (* (* (sin x) (sqrt 2.0)) t_0) 2.0))
        (t_2 (+ 3.0 (sqrt 5.0))))
   (if (<= x -0.0025)
     (/
      t_1
      (*
       (fma 2.0 (/ (cos y) t_2) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
       3.0))
     (if (<= x 0.0038)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (- 1.0 (cos y)) t_0) (sqrt 2.0))
         2.0)
        (fma
         (/ 6.0 t_2)
         (cos y)
         (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 3.0)))
       (/
        t_1
        (fma
         1.5
         (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(x), -0.0625, sin(y));
	double t_1 = fma((cos(x) - cos(y)), ((sin(x) * sqrt(2.0)) * t_0), 2.0);
	double t_2 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -0.0025) {
		tmp = t_1 / (fma(2.0, (cos(y) / t_2), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else if (x <= 0.0038) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (((1.0 - cos(y)) * t_0) * sqrt(2.0)), 2.0) / fma((6.0 / t_2), cos(y), (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0));
	} else {
		tmp = t_1 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(x), -0.0625, sin(y))
	t_1 = fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(x) * sqrt(2.0)) * t_0), 2.0)
	t_2 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0025)
		tmp = Float64(t_1 / Float64(fma(2.0, Float64(cos(y) / t_2), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	elseif (x <= 0.0038)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * t_0) * sqrt(2.0)), 2.0) / fma(Float64(6.0 / t_2), cos(y), Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0)));
	else
		tmp = Float64(t_1 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0025], N[(t$95$1 / N[(N[(2.0 * N[(N[Cos[y], $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0038], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00250000000000000005

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.9

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

   7. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   9. Applied rewrites 66.0%

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

   if -0.00250000000000000005 < x < 0.00379999999999999999

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 99.7

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

   11. Applied rewrites 99.7%

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

   if 0.00379999999999999999 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 74.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0025:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(2, \frac{\cos y}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)\\ t_1 := 3 + \sqrt{5}\\ \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(2, \frac{\cos y}{t\_1}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{t\_1}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (- (cos x) (cos y))
          (* (* (sin x) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y)))
          2.0))
        (t_1 (+ 3.0 (sqrt 5.0))))
   (if (<= x -0.0023)
     (/
      t_0
      (*
       (fma 2.0 (/ (cos y) t_1) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
       3.0))
     (if (<= x 0.0038)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (- 1.0 (cos y)) (sin y)) (sqrt 2.0))
         2.0)
        (fma
         (/ 6.0 t_1)
         (cos y)
         (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 3.0)))
       (/
        t_0
        (fma
         1.5
         (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((cos(x) - cos(y)), ((sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0);
	double t_1 = 3.0 + sqrt(5.0);
	double tmp;
	if (x <= -0.0023) {
		tmp = t_0 / (fma(2.0, (cos(y) / t_1), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else if (x <= 0.0038) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (((1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma((6.0 / t_1), cos(y), (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0));
	} else {
		tmp = t_0 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0)
	t_1 = Float64(3.0 + sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0023)
		tmp = Float64(t_0 / Float64(fma(2.0, Float64(cos(y) / t_1), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	elseif (x <= 0.0038)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma(Float64(6.0 / t_1), cos(y), Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0)));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0023], N[(t$95$0 / N[(N[(2.0 * N[(N[Cos[y], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0038], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0023

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.9

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

   7. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   9. Applied rewrites 66.0%

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

   if -0.0023 < x < 0.00379999999999999999

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-sin.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.00379999999999999999 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 74.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(2, \frac{\cos y}{3 + \sqrt{5}}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot t\_1 + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (- (cos x) (cos y))
          (* (* (sin x) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y)))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0023)
     (/
      t_0
      (*
       (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (+ (* (* 0.5 (cos y)) t_1) 1.0))
       3.0))
     (if (<= x 0.0038)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (- 1.0 (cos y)) (sin y)) (sqrt 2.0))
         2.0)
        (fma
         (/ 6.0 (+ 3.0 (sqrt 5.0)))
         (cos y)
         (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 3.0)))
       (/
        t_0
        (fma 1.5 (fma (cos y) t_1 (* (- (sqrt 5.0) 1.0) (cos x))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((cos(x) - cos(y)), ((sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0023) {
		tmp = t_0 / (fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (((0.5 * cos(y)) * t_1) + 1.0)) * 3.0);
	} else if (x <= 0.0038) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (((1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0));
	} else {
		tmp = t_0 / fma(1.5, fma(cos(y), t_1, ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0023)
		tmp = Float64(t_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));
	elseif (x <= 0.0038)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0)));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(y), t_1, Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0023], N[(t$95$0 / 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], If[LessEqual[x, 0.0038], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0023

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.9

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

   7. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 65.9

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

   9. Applied rewrites 65.9%

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

   if -0.0023 < x < 0.00379999999999999999

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-sin.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.00379999999999999999 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 74.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(t\_1 \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (- (cos x) (cos y))
          (* (* (sin x) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y)))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0023)
     (/
      t_0
      (*
       (fma (* t_1 (cos y)) 0.5 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
       3.0))
     (if (<= x 0.0038)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (- 1.0 (cos y)) (sin y)) (sqrt 2.0))
         2.0)
        (fma
         (/ 6.0 (+ 3.0 (sqrt 5.0)))
         (cos y)
         (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 3.0)))
       (/
        t_0
        (fma 1.5 (fma (cos y) t_1 (* (- (sqrt 5.0) 1.0) (cos x))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((cos(x) - cos(y)), ((sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0023) {
		tmp = t_0 / (fma((t_1 * cos(y)), 0.5, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else if (x <= 0.0038) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (((1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0));
	} else {
		tmp = t_0 / fma(1.5, fma(cos(y), t_1, ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0023)
		tmp = Float64(t_0 / Float64(fma(Float64(t_1 * cos(y)), 0.5, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	elseif (x <= 0.0038)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0)));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(y), t_1, Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0023], N[(t$95$0 / N[(N[(N[(t$95$1 * 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], If[LessEqual[x, 0.0038], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0023

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.9

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

   7. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 65.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 65.9

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

   9. Applied rewrites 65.9%

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

   if -0.0023 < x < 0.00379999999999999999

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-sin.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.00379999999999999999 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 74.5

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

   5. Applied rewrites 74.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.5

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 74.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y)))
           2.0)
          (fma
           1.5
           (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
           3.0))))
   (if (<= x -0.0023)
     t_0
     (if (<= x 0.0038)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (* (- 1.0 (cos y)) (sin y)) (sqrt 2.0))
         2.0)
        (fma
         (/ 6.0 (+ 3.0 (sqrt 5.0)))
         (cos y)
         (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 3.0)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((cos(x) - cos(y)), ((sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	double tmp;
	if (x <= -0.0023) {
		tmp = t_0;
	} else if (x <= 0.0038) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (((1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0))
	tmp = 0.0
	if (x <= -0.0023)
		tmp = t_0;
	elseif (x <= 0.0038)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(Float64(1.0 - cos(y)) * sin(y)) * sqrt(2.0)), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0023], t$95$0, If[LessEqual[x, 0.0038], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0023 or 0.00379999999999999999 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 69.5

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

   5. Applied rewrites 69.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.5

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

   7. Applied rewrites 69.5%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 69.6%

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

   if -0.0023 < x < 0.00379999999999999999

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-sin.f64 99.6

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

   11. Applied rewrites 99.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.0038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ t_2 := \mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_1 \cdot 3\right)\\ t_3 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.0125:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(t\_3 \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{t\_2}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(\mathsf{fma}\left(\sin x, -0.0625, y\right) \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, t\_3 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_1, 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))
        (t_2 (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_1 3.0)))
        (t_3 (- 1.0 (cos y))))
   (if (<= y -0.0125)
     (/ (fma t_0 (* (* t_3 (sin y)) (sqrt 2.0)) 2.0) t_2)
     (if (<= y 2.2e-13)
       (/
        (fma
         t_0
         (* (* (fma (sin x) -0.0625 y) (- (cos x) 1.0)) (sqrt 2.0))
         2.0)
        t_2)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* t_3 (sqrt 2.0)) 2.0)
        (fma t_1 3.0 (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double t_2 = fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_1 * 3.0));
	double t_3 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.0125) {
		tmp = fma(t_0, ((t_3 * sin(y)) * sqrt(2.0)), 2.0) / t_2;
	} else if (y <= 2.2e-13) {
		tmp = fma(t_0, ((fma(sin(x), -0.0625, y) * (cos(x) - 1.0)) * sqrt(2.0)), 2.0) / t_2;
	} else {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), (t_3 * sqrt(2.0)), 2.0) / fma(t_1, 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	t_2 = fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_1 * 3.0))
	t_3 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.0125)
		tmp = Float64(fma(t_0, Float64(Float64(t_3 * sin(y)) * sqrt(2.0)), 2.0) / t_2);
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(t_0, Float64(Float64(fma(sin(x), -0.0625, y) * Float64(cos(x) - 1.0)) * sqrt(2.0)), 2.0) / t_2);
	else
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(t_3 * sqrt(2.0)), 2.0) / fma(t_1, 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$1 * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0125], N[(N[(t$95$0 * N[(N[(t$95$3 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 2.2e-13], N[(N[(t$95$0 * N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$3 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$1 * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.012500000000000001

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.4

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

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 N/A

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

       5. lower-sin.f64 70.4

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

   11. Applied rewrites 70.4%

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

   if -0.012500000000000001 < y < 2.19999999999999997e-13

   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

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

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

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

       8. distribute-rgt-out N/A

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

       9. +-commutative N/A

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

       10. sub-neg N/A

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

       11. lower-*.f64 N/A

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

       12. lower--.f64 N/A

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

       13. lower-cos.f64 N/A

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

       14. sub-neg N/A

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

       15. +-commutative N/A

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

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

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

       17. metadata-eval N/A

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

       18. *-commutative N/A

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

   11. Applied rewrites 99.6%

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

   if 2.19999999999999997e-13 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 64.2

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

   7. Applied rewrites 64.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0125:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\mathsf{fma}\left(\sin x, -0.0625, y\right) \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 - \cos y\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(t\_2 \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_0 \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, t\_2 \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_0, 3, \left(t\_1 \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- 1.0 (cos y))))
   (if (<= y -7.2e-8)
     (/
      (fma (fma (sin y) -0.0625 (sin x)) (* (* t_2 (sin y)) (sqrt 2.0)) 2.0)
      (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_0 3.0)))
     (if (<= y 2.2e-13)
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (sin x) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y)))
         2.0)
        (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_1) 3.0))
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* t_2 (sqrt 2.0)) 2.0)
        (fma t_0 3.0 (* (* t_1 0.5) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 - cos(y);
	double tmp;
	if (y <= -7.2e-8) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), ((t_2 * sin(y)) * sqrt(2.0)), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_0 * 3.0));
	} else if (y <= 2.2e-13) {
		tmp = fma((cos(x) - cos(y)), ((sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), t_1), 3.0);
	} else {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), (t_2 * sqrt(2.0)), 2.0) / fma(t_0, 3.0, ((t_1 * 0.5) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -7.2e-8)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(Float64(t_2 * sin(y)) * sqrt(2.0)), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_0 * 3.0)));
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_1), 3.0));
	else
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(t_2 * sqrt(2.0)), 2.0) / fma(t_0, 3.0, Float64(Float64(t_1 * 0.5) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e-8], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-13], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(t$95$2 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(t$95$1 * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999962e-8

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \color{blue}{\left(\sin y \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

       3. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

       4. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right), \sqrt{2} \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

       5. lower-sin.f64 70.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sin y}\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   11. Applied rewrites 70.9%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sin y\right)}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   if -7.19999999999999962e-8 < y < 2.19999999999999997e-13

   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. 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. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 99.4

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-sqrt.f64 64.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \left(\left(1 - \cos y\right) \cdot \sin y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := {\sin y}^{2} \cdot -0.0625\\ t_2 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(t\_1 \cdot \sqrt{2}\right) \cdot t\_0 + 2}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_2 \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_3\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_2, 3, \left(t\_3 \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* (pow (sin y) 2.0) -0.0625))
        (t_2 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -7.2e-8)
     (/
      (+ (* (* t_1 (sqrt 2.0)) t_0) 2.0)
      (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_2 3.0)))
     (if (<= y 2.2e-13)
       (/
        (fma t_0 (* (* (sin x) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y))) 2.0)
        (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_3) 3.0))
       (/
        (fma t_1 (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma t_2 3.0 (* (* t_3 0.5) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = pow(sin(y), 2.0) * -0.0625;
	double t_2 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -7.2e-8) {
		tmp = (((t_1 * sqrt(2.0)) * t_0) + 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_2 * 3.0));
	} else if (y <= 2.2e-13) {
		tmp = fma(t_0, ((sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), t_3), 3.0);
	} else {
		tmp = fma(t_1, ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(t_2, 3.0, ((t_3 * 0.5) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64((sin(y) ^ 2.0) * -0.0625)
	t_2 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -7.2e-8)
		tmp = Float64(Float64(Float64(Float64(t_1 * sqrt(2.0)) * t_0) + 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_2 * 3.0)));
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(t_0, Float64(Float64(sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_3), 3.0));
	else
		tmp = Float64(fma(t_1, Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(t_2, 3.0, Float64(Float64(t_3 * 0.5) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e-8], N[(N[(N[(N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$2 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-13], N[(N[(t$95$0 * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$2 * 3.0 + N[(N[(t$95$3 * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999962e-8

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lower-sqrt.f64 70.4

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   9. Applied rewrites 70.4%

      \[\leadsto \frac{2 + \color{blue}{\left(\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}\right)} \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   if -7.19999999999999962e-8 < y < 2.19999999999999997e-13

   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. 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. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 99.4

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-sqrt.f64 64.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_1 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_1 \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_2\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(t\_1, 3, \left(t\_2 \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (pow (sin y) 2.0) -0.0625)
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_1 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -7.2e-8)
     (/ t_0 (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_1 3.0)))
     (if (<= y 2.2e-13)
       (/
        (fma
         (- (cos x) (cos y))
         (* (* (sin x) (sqrt 2.0)) (fma (sin x) -0.0625 (sin y)))
         2.0)
        (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_2) 3.0))
       (/ t_0 (fma t_1 3.0 (* (* t_2 0.5) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_1 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -7.2e-8) {
		tmp = t_0 / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_1 * 3.0));
	} else if (y <= 2.2e-13) {
		tmp = fma((cos(x) - cos(y)), ((sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), t_2), 3.0);
	} else {
		tmp = t_0 / fma(t_1, 3.0, ((t_2 * 0.5) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_1 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -7.2e-8)
		tmp = Float64(t_0 / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_1 * 3.0)));
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(Float64(cos(x) - cos(y)), Float64(Float64(sin(x) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_2), 3.0));
	else
		tmp = Float64(t_0 / fma(t_1, 3.0, Float64(Float64(t_2 * 0.5) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e-8], N[(t$95$0 / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$1 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-13], N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(t$95$1 * 3.0 + N[(N[(t$95$2 * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999962e-8

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lower-sqrt.f64 70.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   9. Applied rewrites 70.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   if -7.19999999999999962e-8 < y < 2.19999999999999997e-13

   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. 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. *-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lower-sin.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 99.4

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-sqrt.f64 64.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \left(\sin x \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_1 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ t_2 := \mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_1 \cdot 3\right)\\ \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{t\_0}{t\_2}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(t\_1, 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (pow (sin y) 2.0) -0.0625)
          (* (- 1.0 (cos y)) (sqrt 2.0))
          2.0))
        (t_1 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))
        (t_2 (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_1 3.0))))
   (if (<= y -0.0009)
     (/ t_0 t_2)
     (if (<= y 2.2e-13)
       (/
        (fma (* (fma (cos x) -0.0625 0.0625) (pow (sin x) 2.0)) (sqrt 2.0) 2.0)
        t_2)
       (/ t_0 (fma t_1 3.0 (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0);
	double t_1 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double t_2 = fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_1 * 3.0));
	double tmp;
	if (y <= -0.0009) {
		tmp = t_0 / t_2;
	} else if (y <= 2.2e-13) {
		tmp = fma((fma(cos(x), -0.0625, 0.0625) * pow(sin(x), 2.0)), sqrt(2.0), 2.0) / t_2;
	} else {
		tmp = t_0 / fma(t_1, 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_1 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	t_2 = fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_1 * 3.0))
	tmp = 0.0
	if (y <= -0.0009)
		tmp = Float64(t_0 / t_2);
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * (sin(x) ^ 2.0)), sqrt(2.0), 2.0) / t_2);
	else
		tmp = Float64(t_0 / fma(t_1, 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$1 * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0009], N[(t$95$0 / t$95$2), $MachinePrecision], If[LessEqual[y, 2.2e-13], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$0 / N[(t$95$1 * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999998e-4

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lower-sqrt.f64 69.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   9. Applied rewrites 69.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   if -8.9999999999999998e-4 < y < 2.19999999999999997e-13

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

   9. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   if 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-sqrt.f64 64.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 64.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\\ t_1 := \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_0, 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_0 \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))
        (t_1
         (/
          (fma
           (* (pow (sin y) 2.0) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (fma t_0 3.0 (* (* (- 3.0 (sqrt 5.0)) 0.5) (* 3.0 (cos y)))))))
   (if (<= y -0.0009)
     t_1
     (if (<= y 2.2e-13)
       (/
        (fma (* (fma (cos x) -0.0625 0.0625) (pow (sin x) 2.0)) (sqrt 2.0) 2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* t_0 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0);
	double t_1 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(t_0, 3.0, (((3.0 - sqrt(5.0)) * 0.5) * (3.0 * cos(y))));
	double tmp;
	if (y <= -0.0009) {
		tmp = t_1;
	} else if (y <= 2.2e-13) {
		tmp = fma((fma(cos(x), -0.0625, 0.0625) * pow(sin(x), 2.0)), sqrt(2.0), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (t_0 * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)
	t_1 = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(t_0, 3.0, Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * Float64(3.0 * cos(y)))))
	tmp = 0.0
	if (y <= -0.0009)
		tmp = t_1;
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * (sin(x) ^ 2.0)), sqrt(2.0), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(t_0 * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0009], t$95$1, If[LessEqual[y, 2.2e-13], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(t$95$0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999998e-4 or 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-sqrt.f64 66.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 66.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -8.9999999999999998e-4 < y < 2.19999999999999997e-13

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

   9. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\\ t_1 := \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot 3, \cos y, t\_0\right)}\\ \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0) 3.0))
        (t_1
         (/
          (fma
           (* (pow (sin y) 2.0) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (fma (* (* (- 3.0 (sqrt 5.0)) 0.5) 3.0) (cos y) t_0))))
   (if (<= y -0.0009)
     t_1
     (if (<= y 2.2e-13)
       (/
        (fma (* (fma (cos x) -0.0625 0.0625) (pow (sin x) 2.0)) (sqrt 2.0) 2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) t_0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0;
	double t_1 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((((3.0 - sqrt(5.0)) * 0.5) * 3.0), cos(y), t_0);
	double tmp;
	if (y <= -0.0009) {
		tmp = t_1;
	} else if (y <= 2.2e-13) {
		tmp = fma((fma(cos(x), -0.0625, 0.0625) * pow(sin(x), 2.0)), sqrt(2.0), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0) * 3.0)
	t_1 = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(Float64(Float64(3.0 - sqrt(5.0)) * 0.5) * 3.0), cos(y), t_0))
	tmp = 0.0
	if (y <= -0.0009)
		tmp = t_1;
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * (sin(x) ^ 2.0)), sqrt(2.0), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 3.0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0009], t$95$1, If[LessEqual[y, 2.2e-13], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999998e-4 or 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot \frac{-1}{16}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. lower-sqrt.f64 66.5

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Applied rewrites 66.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   if -8.9999999999999998e-4 < y < 2.19999999999999997e-13

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

   9. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot 3, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(\left(3 - \sqrt{5}\right) \cdot 0.5\right) \cdot 3, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(t\_0, \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, t\_0, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (/
          (fma
           (* (pow (sin y) 2.0) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           (fma t_0 (cos x) (+ (* (* 0.5 (cos y)) (- 3.0 (sqrt 5.0))) 1.0))
           3.0))))
   (if (<= y -0.0009)
     t_1
     (if (<= y 2.2e-13)
       (/
        (fma (* (fma (cos x) -0.0625 0.0625) (pow (sin x) 2.0)) (sqrt 2.0) 2.0)
        (fma (/ 6.0 (+ 3.0 (sqrt 5.0))) (cos y) (* (fma (cos x) t_0 1.0) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(t_0, cos(x), (((0.5 * cos(y)) * (3.0 - sqrt(5.0))) + 1.0)) * 3.0);
	double tmp;
	if (y <= -0.0009) {
		tmp = t_1;
	} else if (y <= 2.2e-13) {
		tmp = fma((fma(cos(x), -0.0625, 0.0625) * pow(sin(x), 2.0)), sqrt(2.0), 2.0) / fma((6.0 / (3.0 + sqrt(5.0))), cos(y), (fma(cos(x), t_0, 1.0) * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(t_0, cos(x), Float64(Float64(Float64(0.5 * cos(y)) * Float64(3.0 - sqrt(5.0))) + 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -0.0009)
		tmp = t_1;
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * (sin(x) ^ 2.0)), sqrt(2.0), 2.0) / fma(Float64(6.0 / Float64(3.0 + sqrt(5.0))), cos(y), Float64(fma(cos(x), t_0, 1.0) * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0009], t$95$1, If[LessEqual[y, 2.2e-13], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999998e-4 or 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 27.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      6. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   7. Applied rewrites 27.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-sqrt.f64 66.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 66.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -8.9999999999999998e-4 < y < 2.19999999999999997e-13

   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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

   9. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\frac{6}{\sqrt{5} + 3}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0009:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{6}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot t\_0 + 1\right) \cdot 3}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (pow (sin y) 2.0) -0.0625)
           (* (- 1.0 (cos y)) (sqrt 2.0))
           2.0)
          (*
           (fma
            (fma (sqrt 5.0) 0.5 -0.5)
            (cos x)
            (+ (* (* 0.5 (cos y)) t_0) 1.0))
           3.0))))
   (if (<= y -7.2e-8)
     t_1
     (if (<= y 2.2e-13)
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
        (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (((0.5 * cos(y)) * t_0) + 1.0)) * 3.0);
	double tmp;
	if (y <= -7.2e-8) {
		tmp = t_1;
	} else if (y <= 2.2e-13) {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), t_0), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(Float64(0.5 * cos(y)) * t_0) + 1.0)) * 3.0))
	tmp = 0.0
	if (y <= -7.2e-8)
		tmp = t_1;
	elseif (y <= 2.2e-13)
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e-8], t$95$1, If[LessEqual[y, 2.2e-13], 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[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999962e-8 or 2.19999999999999997e-13 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 27.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      6. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   7. Applied rewrites 27.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

      11. lower-sqrt.f64 66.8

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   10. Applied rewrites 66.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -7.19999999999999962e-8 < y < 2.19999999999999997e-13

   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. 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 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1\right) \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot t\_1\right)}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          (pow (sin x) 2.0)
          2.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -5.8e-7)
     (/
      t_0
      (fma
       (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
       3.0
       (* (* 1.5 (cos y)) t_1)))
     (if (<= x 1.06e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 6.0 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        t_0
        (fma 1.5 (fma (cos y) t_1 (* (- (sqrt 5.0) 1.0) (cos x))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -5.8e-7) {
		tmp = t_0 / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((1.5 * cos(y)) * t_1));
	} else if (x <= 1.06e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((cos(y) / (3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = t_0 / fma(1.5, fma(cos(y), t_1, ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -5.8e-7)
		tmp = Float64(t_0 / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(1.5 * cos(y)) * t_1)));
	elseif (x <= 1.06e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(y), t_1, Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-7], N[(t$95$0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999995e-7

   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 62.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\frac{\sqrt{5} - 1}{2} \cdot \cos x} + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\cos x \cdot \frac{\sqrt{5} - 1}{2}} + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \color{blue}{\frac{\sqrt{5} - 1}{2}} + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \frac{\color{blue}{\sqrt{5} - 1}}{2} + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. div-sub N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \color{blue}{\left(\frac{\sqrt{5}}{2} - \frac{1}{2}\right)} + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} - \frac{1}{2}\right) + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \left(\sqrt{5} \cdot \color{blue}{\frac{1}{2}} - \frac{1}{2}\right) + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \left(\sqrt{5} \cdot \frac{1}{2} - \color{blue}{\frac{1}{2}}\right) + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \left(\sqrt{5} \cdot \frac{1}{2} + \color{blue}{\frac{-1}{2}}\right) + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\cos x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} + 1\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 1.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   if -5.7999999999999995e-7 < x < 1.06e-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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   if 1.06e-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 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 72.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(t\_1 \cdot \cos y, 0.5, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          (pow (sin x) 2.0)
          2.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -5.8e-7)
     (/
      t_0
      (*
       (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) (fma (* t_1 (cos y)) 0.5 1.0))
       3.0))
     (if (<= x 1.06e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 6.0 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        t_0
        (fma 1.5 (fma (cos y) t_1 (* (- (sqrt 5.0) 1.0) (cos x))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -5.8e-7) {
		tmp = t_0 / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), fma((t_1 * cos(y)), 0.5, 1.0)) * 3.0);
	} else if (x <= 1.06e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((cos(y) / (3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = t_0 / fma(1.5, fma(cos(y), t_1, ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -5.8e-7)
		tmp = Float64(t_0 / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), fma(Float64(t_1 * cos(y)), 0.5, 1.0)) * 3.0));
	elseif (x <= 1.06e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(y), t_1, Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-7], N[(t$95$0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999995e-7

   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 62.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      6. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)}\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + 1}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\frac{1}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\frac{1}{2} \cdot \cos y\right)} \cdot \left(3 - \sqrt{5}\right) + 1\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2}} + 1\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\mathsf{fma}\left(\cos y \cdot \left(3 - \sqrt{5}\right), \frac{1}{2}, 1\right)}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}, \frac{1}{2}, 1\right)\right)} \]

      9. lower-*.f64 62.9

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}, 0.5, 1\right)\right)} \]

   9. Applied rewrites 62.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1\right)}\right)} \]

   if -5.7999999999999995e-7 < x < 1.06e-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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   if 1.06e-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 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 72.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(t\_0 \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (pow (sin x) 2.0)))
   (if (<= x -5.8e-7)
     (*
      (fma t_1 (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0)) 2.0)
      (/
       0.3333333333333333
       (fma (* t_0 0.5) (cos y) (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))))
     (if (<= x 1.06e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 6.0 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_1 2.0)
        (fma 1.5 (fma (cos y) t_0 (* (- (sqrt 5.0) 1.0) (cos x))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -5.8e-7) {
		tmp = fma(t_1, (fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), 2.0) * (0.3333333333333333 / fma((t_0 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	} else if (x <= 1.06e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((cos(y) / (3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_1, 2.0) / fma(1.5, fma(cos(y), t_0, ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -5.8e-7)
		tmp = Float64(fma(t_1, Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), 2.0) * Float64(0.3333333333333333 / fma(Float64(t_0 * 0.5), cos(y), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0))));
	elseif (x <= 1.06e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_1, 2.0) / fma(1.5, fma(cos(y), t_0, Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -5.8e-7], N[(N[(t$95$1 * N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[(t$95$0 * 0.5), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999995e-7

   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 62.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      6. div-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \left(\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right) + \color{blue}{\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   9. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, 2\right)} \]

   if -5.7999999999999995e-7 < x < 1.06e-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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   if 1.06e-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 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 72.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left({\sin x}^{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma
           1.5
           (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
           3.0))))
   (if (<= x -5.8e-7)
     t_0
     (if (<= x 1.06e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 6.0 (fma 1.5 (sqrt 5.0) 1.5)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	double tmp;
	if (x <= -5.8e-7) {
		tmp = t_0;
	} else if (x <= 1.06e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((cos(y) / (3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0))
	tmp = 0.0
	if (x <= -5.8e-7)
		tmp = t_0;
	elseif (x <= 1.06e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(-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[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-7], t$95$0, If[LessEqual[x, 1.06e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7999999999999995e-7 or 1.06e-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 66.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 66.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]

   if -5.7999999999999995e-7 < x < 1.06e-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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right)\\ t_1 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot t\_1, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, t\_0, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(1.5, t\_0, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))))
        (t_1 (pow (sin x) 2.0)))
   (if (<= x -5.8e-7)
     (*
      (/
       (fma (* (fma (cos x) -0.0625 0.0625) t_1) (sqrt 2.0) 2.0)
       (fma 0.5 t_0 1.0))
      0.3333333333333333)
     (if (<= x 2.3e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma (/ (cos y) (+ 3.0 (sqrt 5.0))) 6.0 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_1 2.0)
        (fma 1.5 t_0 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0)));
	double t_1 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -5.8e-7) {
		tmp = (fma((fma(cos(x), -0.0625, 0.0625) * t_1), sqrt(2.0), 2.0) / fma(0.5, t_0, 1.0)) * 0.3333333333333333;
	} else if (x <= 2.3e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma((cos(y) / (3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_1, 2.0) / fma(1.5, t_0, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0)))
	t_1 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -5.8e-7)
		tmp = Float64(Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * t_1), sqrt(2.0), 2.0) / fma(0.5, t_0, 1.0)) * 0.3333333333333333);
	elseif (x <= 2.3e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(cos(y) / Float64(3.0 + sqrt(5.0))), 6.0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_1, 2.0) / fma(1.5, t_0, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -5.8e-7], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 2.3e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(1.5 * t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999995e-7

   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 62.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

      13. lower-sqrt.f64 22.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

   8. Applied rewrites 22.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 62.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -5.7999999999999995e-7 < x < 2.3e-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

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)\right)}} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y} + 3 \cdot \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(3 \cdot \frac{3 - \sqrt{5}}{2}\right) \cdot \cos y + \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{3 - \sqrt{5}}{2}, \cos y, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}} \cdot \left(3 - \sqrt{5}\right), \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. flip-- N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{\frac{3}{2}} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\frac{3}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{3 + 3}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 + 3}{3 + \sqrt{5}}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{\color{blue}{6}}{3 + \sqrt{5}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      19. +-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      20. lower-+.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{6}{\color{blue}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{6}{\sqrt{5} + 3}}, \cos y, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\color{blue}{6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{\sqrt{5} + 3}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   if 2.3e-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 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 71.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{\cos y}{3 + \sqrt{5}}, 6, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(t\_0, \cos x, t\_1\right)\\ t_3 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot t\_3, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, t\_2, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_3, 2\right)}{\mathsf{fma}\left(1.5, t\_2, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma t_0 (cos x) t_1))
        (t_3 (pow (sin x) 2.0)))
   (if (<= x -5.8e-7)
     (*
      (/
       (fma (* (fma (cos x) -0.0625 0.0625) t_3) (sqrt 2.0) 2.0)
       (fma 0.5 t_2 1.0))
      0.3333333333333333)
     (if (<= x 2.3e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_3 2.0)
        (fma 1.5 t_2 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(t_0, cos(x), t_1);
	double t_3 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -5.8e-7) {
		tmp = (fma((fma(cos(x), -0.0625, 0.0625) * t_3), sqrt(2.0), 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333;
	} else if (x <= 2.3e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_3, 2.0) / fma(1.5, t_2, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(t_0, cos(x), t_1)
	t_3 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -5.8e-7)
		tmp = Float64(Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * t_3), sqrt(2.0), 2.0) / fma(0.5, t_2, 1.0)) * 0.3333333333333333);
	elseif (x <= 2.3e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_3, 2.0) / fma(1.5, t_2, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -5.8e-7], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 2.3e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(1.5 * t$95$2 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999995e-7

   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 62.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

      13. lower-sqrt.f64 22.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

   8. Applied rewrites 22.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 62.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -5.7999999999999995e-7 < x < 2.3e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      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 63.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

      13. lower-sqrt.f64 63.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.4

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   if 2.3e-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 72.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 71.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot {\sin x}^{2}, \sqrt{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma t_0 (cos x) t_1) 3.0))))
   (if (<= x -5.8e-7)
     t_2
     (if (<= x 2.3e-5)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	double tmp;
	if (x <= -5.8e-7) {
		tmp = t_2;
	} else if (x <= 2.3e-5) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0))
	tmp = 0.0
	if (x <= -5.8e-7)
		tmp = t_2;
	elseif (x <= 2.3e-5)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-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[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-7], t$95$2, If[LessEqual[x, 2.3e-5], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7999999999999995e-7 or 2.3e-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 66.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 66.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]

   if -5.7999999999999995e-7 < x < 2.3e-5

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      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 63.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

      13. lower-sqrt.f64 63.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

   8. Applied rewrites 63.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

       11. lower-sqrt.f64 99.4

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   11. Applied rewrites 99.4%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
  (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-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[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 64.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 62.9%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]
9. Add Preprocessing

Alternative 29: 45.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   1.5
   (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 64.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

   13. lower-sqrt.f64 43.1

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

12. Taylor expanded in y around inf

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 45.6%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}} \]
15. Add Preprocessing

Alternative 30: 42.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 64.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

   13. lower-sqrt.f64 43.1

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 43.5%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}} \]
15. Add Preprocessing

Alternative 31: 42.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (- (sqrt 5.0) 1.0)) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) - 1.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) - 1.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 64.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

   13. lower-sqrt.f64 43.1

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
12. Add Preprocessing

Alternative 32: 40.2% accurate, 78.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{6} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 2.0 6.0))

C

double code(double x, double y) {
	return 2.0 / 6.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 / 6.0d0
end function

Java

public static double code(double x, double y) {
	return 2.0 / 6.0;
}

Python

def code(x, y):
	return 2.0 / 6.0

Julia

function code(x, y)
	return Float64(2.0 / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 / 6.0;
end

Wolfram

code[x_, y_] := N[(2.0 / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 64.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

   13. lower-sqrt.f64 43.1

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

8. Applied rewrites 43.1%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 43.1%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{2}{6} \]
13. Step-by-step derivation

14. Applied rewrites 41.1%

    \[\leadsto \frac{2}{6} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024273 
(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))))))