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

Percentage Accurate: 99.3% → 99.3%

Time: 18.6s

Alternatives: 29

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

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))), 2.0) / fma(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)), 1.5, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 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[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $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-rgt-in N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 99.3

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

4. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

6. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $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. 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 x around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-sqrt.f64 57.5

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

7. Applied rewrites 57.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. sub-neg N/A

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

   12. distribute-rgt-in N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-cos.f64 60.2

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

5. Applied rewrites 60.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 31.3%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

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

Alternative 4: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, 1\right), 3, \left(0.5 \cdot t\_0\right) \cdot \left(3 \cdot \cos y\right)\right)\\ t_3 := \sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot t\_3, 2\right)}{\mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, \mathsf{fma}\left(t\_1, \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{2 - \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(y \cdot y, 0.5, \cos x - 1\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot t\_3\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (fma (fma (cos x) t_1 1.0) 3.0 (* (* 0.5 t_0) (* 3.0 (cos y)))))
        (t_3 (* (sqrt 2.0) (fma (sin y) -0.0625 (sin x)))))
   (if (<= y -0.06)
     (/
      (fma (sin y) (* (- (cos x) (cos y)) t_3) 2.0)
      (fma (* t_0 (cos y)) 1.5 (* (fma t_1 (cos x) 1.0) 3.0)))
     (if (<= y 0.0029)
       (/
        (-
         2.0
         (*
          (*
           (- (/ (sin x) 16.0) (sin y))
           (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0)))
          (fma (* y y) 0.5 (- (cos x) 1.0))))
        t_2)
       (/ (- 2.0 (* (- (cos y) (cos x)) (* (sin y) t_3))) t_2)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = fma(fma(cos(x), t_1, 1.0), 3.0, ((0.5 * t_0) * (3.0 * cos(y))));
	double t_3 = sqrt(2.0) * fma(sin(y), -0.0625, sin(x));
	double tmp;
	if (y <= -0.06) {
		tmp = fma(sin(y), ((cos(x) - cos(y)) * t_3), 2.0) / fma((t_0 * cos(y)), 1.5, (fma(t_1, cos(x), 1.0) * 3.0));
	} else if (y <= 0.0029) {
		tmp = (2.0 - ((((sin(x) / 16.0) - sin(y)) * ((sin(x) - (sin(y) / 16.0)) * sqrt(2.0))) * fma((y * y), 0.5, (cos(x) - 1.0)))) / t_2;
	} else {
		tmp = (2.0 - ((cos(y) - cos(x)) * (sin(y) * t_3))) / t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = fma(fma(cos(x), t_1, 1.0), 3.0, Float64(Float64(0.5 * t_0) * Float64(3.0 * cos(y))))
	t_3 = Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))
	tmp = 0.0
	if (y <= -0.06)
		tmp = Float64(fma(sin(y), Float64(Float64(cos(x) - cos(y)) * t_3), 2.0) / fma(Float64(t_0 * cos(y)), 1.5, Float64(fma(t_1, cos(x), 1.0) * 3.0)));
	elseif (y <= 0.0029)
		tmp = Float64(Float64(2.0 - Float64(Float64(Float64(Float64(sin(x) / 16.0) - sin(y)) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0))) * fma(Float64(y * y), 0.5, Float64(cos(x) - 1.0)))) / t_2);
	else
		tmp = Float64(Float64(2.0 - Float64(Float64(cos(y) - cos(x)) * Float64(sin(y) * t_3))) / t_2);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.06], N[(N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0029], N[(N[(2.0 - N[(N[(N[(N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision] - N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.5 + N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 - N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.059999999999999998

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if -0.059999999999999998 < y < 0.0029

   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. 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 99.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 0.0029 < 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 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   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. lower-+.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{2 - \left(\left(\frac{\sin x}{16} - \sin y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(y \cdot y, 0.5, \cos x - 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, \mathsf{fma}\left(t\_1, \cos x, 1\right) \cdot 3\right)\\ t_3 := \sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot t\_3, 2\right)}{t\_2}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(y \cdot y, 0.5, \cos x\right) - 1\right) \cdot t\_3, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot t\_3\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, 1\right), 3, \left(0.5 \cdot t\_0\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (fma (* t_0 (cos y)) 1.5 (* (fma t_1 (cos x) 1.0) 3.0)))
        (t_3 (* (sqrt 2.0) (fma (sin y) -0.0625 (sin x)))))
   (if (<= y -0.06)
     (/ (fma (sin y) (* (- (cos x) (cos y)) t_3) 2.0) t_2)
     (if (<= y 0.0029)
       (/
        (fma
         (fma (sin x) -0.0625 (sin y))
         (* (- (fma (* y y) 0.5 (cos x)) 1.0) t_3)
         2.0)
        t_2)
       (/
        (- 2.0 (* (- (cos y) (cos x)) (* (sin y) t_3)))
        (fma (fma (cos x) t_1 1.0) 3.0 (* (* 0.5 t_0) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = fma((t_0 * cos(y)), 1.5, (fma(t_1, cos(x), 1.0) * 3.0));
	double t_3 = sqrt(2.0) * fma(sin(y), -0.0625, sin(x));
	double tmp;
	if (y <= -0.06) {
		tmp = fma(sin(y), ((cos(x) - cos(y)) * t_3), 2.0) / t_2;
	} else if (y <= 0.0029) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), ((fma((y * y), 0.5, cos(x)) - 1.0) * t_3), 2.0) / t_2;
	} else {
		tmp = (2.0 - ((cos(y) - cos(x)) * (sin(y) * t_3))) / fma(fma(cos(x), t_1, 1.0), 3.0, ((0.5 * t_0) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = fma(Float64(t_0 * cos(y)), 1.5, Float64(fma(t_1, cos(x), 1.0) * 3.0))
	t_3 = Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))
	tmp = 0.0
	if (y <= -0.06)
		tmp = Float64(fma(sin(y), Float64(Float64(cos(x) - cos(y)) * t_3), 2.0) / t_2);
	elseif (y <= 0.0029)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(fma(Float64(y * y), 0.5, cos(x)) - 1.0) * t_3), 2.0) / t_2);
	else
		tmp = Float64(Float64(2.0 - Float64(Float64(cos(y) - cos(x)) * Float64(sin(y) * t_3))) / fma(fma(cos(x), t_1, 1.0), 3.0, Float64(Float64(0.5 * t_0) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.06], N[(N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 0.0029], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.5 + N[Cos[x], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * t$95$3), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 - N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.059999999999999998

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if -0.059999999999999998 < y < 0.0029

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around 0

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-cos.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 0.0029 < 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 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   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. lower-+.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(y \cdot y, 0.5, \cos x\right) - 1\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_3 := \left(\cos x - \cos y\right) \cdot t\_2\\ \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, t\_3, 2\right)}{\mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, \mathsf{fma}\left(t\_1, \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), t\_3, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \mathsf{fma}\left(-0.75, y \cdot y, 1.5\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot t\_2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, 1\right), 3, \left(0.5 \cdot t\_0\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (* (sqrt 2.0) (fma (sin y) -0.0625 (sin x))))
        (t_3 (* (- (cos x) (cos y)) t_2)))
   (if (<= y -0.03)
     (/
      (fma (sin y) t_3 2.0)
      (fma (* t_0 (cos y)) 1.5 (* (fma t_1 (cos x) 1.0) 3.0)))
     (if (<= y 0.0029)
       (/
        (fma (fma (sin x) -0.0625 (sin y)) t_3 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (* (fma -0.75 (* y y) 1.5) t_0)))
       (/
        (- 2.0 (* (- (cos y) (cos x)) (* (sin y) t_2)))
        (fma (fma (cos x) t_1 1.0) 3.0 (* (* 0.5 t_0) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = sqrt(2.0) * fma(sin(y), -0.0625, sin(x));
	double t_3 = (cos(x) - cos(y)) * t_2;
	double tmp;
	if (y <= -0.03) {
		tmp = fma(sin(y), t_3, 2.0) / fma((t_0 * cos(y)), 1.5, (fma(t_1, cos(x), 1.0) * 3.0));
	} else if (y <= 0.0029) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), t_3, 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (fma(-0.75, (y * y), 1.5) * t_0));
	} else {
		tmp = (2.0 - ((cos(y) - cos(x)) * (sin(y) * t_2))) / fma(fma(cos(x), t_1, 1.0), 3.0, ((0.5 * t_0) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))
	t_3 = Float64(Float64(cos(x) - cos(y)) * t_2)
	tmp = 0.0
	if (y <= -0.03)
		tmp = Float64(fma(sin(y), t_3, 2.0) / fma(Float64(t_0 * cos(y)), 1.5, Float64(fma(t_1, cos(x), 1.0) * 3.0)));
	elseif (y <= 0.0029)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), t_3, 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(fma(-0.75, Float64(y * y), 1.5) * t_0)));
	else
		tmp = Float64(Float64(2.0 - Float64(Float64(cos(y) - cos(x)) * Float64(sin(y) * t_2))) / fma(fma(cos(x), t_1, 1.0), 3.0, Float64(Float64(0.5 * t_0) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[y, -0.03], N[(N[(N[Sin[y], $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0029], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$3 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(-0.75 * N[(y * y), $MachinePrecision] + 1.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.029999999999999999

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if -0.029999999999999999 < y < 0.0029

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

   9. Applied rewrites 99.5%

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

   if 0.0029 < 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 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   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. lower-+.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.03:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \mathsf{fma}\left(-0.75, y \cdot y, 1.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_3 := \mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, \mathsf{fma}\left(t\_2, \cos x, 1\right) \cdot 3\right)\\ t_4 := \sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, t\_1 \cdot t\_4, 2\right)}{t\_3}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot t\_1, 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot t\_4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(0.5 \cdot t\_0\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5))
        (t_3 (fma (* t_0 (cos y)) 1.5 (* (fma t_2 (cos x) 1.0) 3.0)))
        (t_4 (* (sqrt 2.0) (fma (sin y) -0.0625 (sin x)))))
   (if (<= y -0.06)
     (/ (fma (sin y) (* t_1 t_4) 2.0) t_3)
     (if (<= y 0.0029)
       (/
        (fma
         (fma (sin x) -0.0625 (sin y))
         (* (* (fma -0.0625 y (sin x)) (sqrt 2.0)) t_1)
         2.0)
        t_3)
       (/
        (- 2.0 (* (- (cos y) (cos x)) (* (sin y) t_4)))
        (fma (fma (cos x) t_2 1.0) 3.0 (* (* 0.5 t_0) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double t_3 = fma((t_0 * cos(y)), 1.5, (fma(t_2, cos(x), 1.0) * 3.0));
	double t_4 = sqrt(2.0) * fma(sin(y), -0.0625, sin(x));
	double tmp;
	if (y <= -0.06) {
		tmp = fma(sin(y), (t_1 * t_4), 2.0) / t_3;
	} else if (y <= 0.0029) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), ((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * t_1), 2.0) / t_3;
	} else {
		tmp = (2.0 - ((cos(y) - cos(x)) * (sin(y) * t_4))) / fma(fma(cos(x), t_2, 1.0), 3.0, ((0.5 * t_0) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	t_3 = fma(Float64(t_0 * cos(y)), 1.5, Float64(fma(t_2, cos(x), 1.0) * 3.0))
	t_4 = Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))
	tmp = 0.0
	if (y <= -0.06)
		tmp = Float64(fma(sin(y), Float64(t_1 * t_4), 2.0) / t_3);
	elseif (y <= 0.0029)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * t_1), 2.0) / t_3);
	else
		tmp = Float64(Float64(2.0 - Float64(Float64(cos(y) - cos(x)) * Float64(sin(y) * t_4))) / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(0.5 * t_0) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.06], N[(N[(N[Sin[y], $MachinePrecision] * N[(t$95$1 * t$95$4), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y, 0.0029], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(2.0 - N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.059999999999999998

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if -0.059999999999999998 < y < 0.0029

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sin.f64 99.4

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

   11. Applied rewrites 99.4%

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

   if 0.0029 < 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 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   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. lower-+.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-sin.f64 61.0

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

   9. Applied rewrites 61.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\sin y \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\mathsf{fma}\left(\sin y, t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot t\_1, 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (- 3.0 (sqrt 5.0)) (cos y))
          1.5
          (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (fma
           (sin y)
           (* t_1 (* (sqrt 2.0) (fma (sin y) -0.0625 (sin x))))
           2.0)
          t_0)))
   (if (<= y -0.06)
     t_2
     (if (<= y 0.0029)
       (/
        (fma
         (fma (sin x) -0.0625 (sin y))
         (* (* (fma -0.0625 y (sin x)) (sqrt 2.0)) t_1)
         2.0)
        t_0)
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(((3.0 - sqrt(5.0)) * cos(y)), 1.5, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_1 = cos(x) - cos(y);
	double t_2 = fma(sin(y), (t_1 * (sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))), 2.0) / t_0;
	double tmp;
	if (y <= -0.06) {
		tmp = t_2;
	} else if (y <= 0.0029) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), ((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * t_1), 2.0) / t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)), 1.5, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(fma(sin(y), Float64(t_1 * Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.06)
		tmp = t_2;
	elseif (y <= 0.0029)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * t_1), 2.0) / t_0);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[y], $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.06], t$95$2, If[LessEqual[y, 0.0029], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.059999999999999998 or 0.0029 < 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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.3

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.059999999999999998 < y < 0.0029

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sin.f64 99.4

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

   11. Applied rewrites 99.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)\\ t_1 := \frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.0132:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (- 3.0 (sqrt 5.0)) (cos y))
          1.5
          (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))
        (t_1
         (/
          (fma
           (sin y)
           (* (- (cos x) (cos y)) (* (sqrt 2.0) (fma (sin y) -0.0625 (sin x))))
           2.0)
          t_0)))
   (if (<= y -0.0132)
     t_1
     (if (<= y 0.0026)
       (/
        (fma
         (fma (sin x) -0.0625 (sin y))
         (* (* (- (cos x) 1.0) (sqrt 2.0)) (fma -0.0625 y (sin x)))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(((3.0 - sqrt(5.0)) * cos(y)), 1.5, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double t_1 = fma(sin(y), ((cos(x) - cos(y)) * (sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))), 2.0) / t_0;
	double tmp;
	if (y <= -0.0132) {
		tmp = t_1;
	} else if (y <= 0.0026) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), (((cos(x) - 1.0) * sqrt(2.0)) * fma(-0.0625, y, sin(x))), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)), 1.5, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0))
	t_1 = Float64(fma(sin(y), Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * fma(sin(y), -0.0625, sin(x)))), 2.0) / t_0)
	tmp = 0.0
	if (y <= -0.0132)
		tmp = t_1;
	elseif (y <= 0.0026)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * fma(-0.0625, y, sin(x))), 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[y], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -0.0132], t$95$1, If[LessEqual[y, 0.0026], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0132 or 0.0025999999999999999 < 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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.3

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-sin.f64 64.2

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

   11. Applied rewrites 64.2%

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

   if -0.0132 < y < 0.0025999999999999999

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-sin.f64 99.1

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

   11. Applied rewrites 99.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0132:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin y, \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_2 := \left(1 - \cos y\right) \cdot \sqrt{2}\\ t_3 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_4 := \mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, \mathsf{fma}\left(t\_3, \cos x, 1\right) \cdot 3\right)\\ \mathbf{if}\;y \leq -0.0132:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_2 \cdot \left(\sin y \cdot -0.0625\right), 2\right)}{t\_4}\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), 2\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, t\_2, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_3, 1\right), 3, \left(0.5 \cdot t\_0\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (sin x) -0.0625 (sin y)))
        (t_2 (* (- 1.0 (cos y)) (sqrt 2.0)))
        (t_3 (fma (sqrt 5.0) 0.5 -0.5))
        (t_4 (fma (* t_0 (cos y)) 1.5 (* (fma t_3 (cos x) 1.0) 3.0))))
   (if (<= y -0.0132)
     (/ (fma t_1 (* t_2 (* (sin y) -0.0625)) 2.0) t_4)
     (if (<= y 0.0026)
       (/
        (fma
         t_1
         (* (* (- (cos x) 1.0) (sqrt 2.0)) (fma -0.0625 y (sin x)))
         2.0)
        t_4)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) t_2 2.0)
        (fma (fma (cos x) t_3 1.0) 3.0 (* (* 0.5 t_0) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sin(x), -0.0625, sin(y));
	double t_2 = (1.0 - cos(y)) * sqrt(2.0);
	double t_3 = fma(sqrt(5.0), 0.5, -0.5);
	double t_4 = fma((t_0 * cos(y)), 1.5, (fma(t_3, cos(x), 1.0) * 3.0));
	double tmp;
	if (y <= -0.0132) {
		tmp = fma(t_1, (t_2 * (sin(y) * -0.0625)), 2.0) / t_4;
	} else if (y <= 0.0026) {
		tmp = fma(t_1, (((cos(x) - 1.0) * sqrt(2.0)) * fma(-0.0625, y, sin(x))), 2.0) / t_4;
	} else {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), t_2, 2.0) / fma(fma(cos(x), t_3, 1.0), 3.0, ((0.5 * t_0) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(sin(x), -0.0625, sin(y))
	t_2 = Float64(Float64(1.0 - cos(y)) * sqrt(2.0))
	t_3 = fma(sqrt(5.0), 0.5, -0.5)
	t_4 = fma(Float64(t_0 * cos(y)), 1.5, Float64(fma(t_3, cos(x), 1.0) * 3.0))
	tmp = 0.0
	if (y <= -0.0132)
		tmp = Float64(fma(t_1, Float64(t_2 * Float64(sin(y) * -0.0625)), 2.0) / t_4);
	elseif (y <= 0.0026)
		tmp = Float64(fma(t_1, Float64(Float64(Float64(cos(x) - 1.0) * sqrt(2.0)) * fma(-0.0625, y, sin(x))), 2.0) / t_4);
	else
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), t_2, 2.0) / fma(fma(cos(x), t_3, 1.0), 3.0, Float64(Float64(0.5 * t_0) * Float64(3.0 * cos(y)))));
	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[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$3 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0132], N[(N[(t$95$1 * N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[y, 0.0026], N[(N[(t$95$1 * N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision] * 3.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0132

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-sqrt.f64 64.7

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

   11. Applied rewrites 64.7%

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

   if -0.0132 < y < 0.0025999999999999999

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-sin.f64 99.1

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

   11. Applied rewrites 99.1%

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

   if 0.0025999999999999999 < 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 99.2%

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

   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. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sqrt.f64 57.1

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

   7. Applied rewrites 57.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0132:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left(\sin y \cdot -0.0625\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.0026:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, y, \sin x\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, \mathsf{fma}\left(t\_1, \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_1, 1\right), 3, \left(0.5 \cdot t\_0\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -8e-6)
     (/
      (fma
       (fma (sin x) -0.0625 (sin y))
       (* (* (sqrt 2.0) (sin x)) (- (cos x) 1.0))
       2.0)
      (fma (* t_0 (cos y)) 1.5 (* (fma t_1 (cos x) 1.0) 3.0)))
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_0 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (-
         2.0
         (* (- (cos y) (cos x)) (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0))))
        (fma (fma (cos x) t_1 1.0) 3.0 (* (* 0.5 t_0) (* 3.0 (cos y)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -8e-6) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), ((sqrt(2.0) * sin(x)) * (cos(x) - 1.0)), 2.0) / fma((t_0 * cos(y)), 1.5, (fma(t_1, cos(x), 1.0) * 3.0));
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = (2.0 - ((cos(y) - cos(x)) * ((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0)))) / fma(fma(cos(x), t_1, 1.0), 3.0, ((0.5 * t_0) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -8e-6)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(sqrt(2.0) * sin(x)) * Float64(cos(x) - 1.0)), 2.0) / fma(Float64(t_0 * cos(y)), 1.5, Float64(fma(t_1, cos(x), 1.0) * 3.0)));
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(Float64(2.0 - Float64(Float64(cos(y) - cos(x)) * Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0)))) / fma(fma(cos(x), t_1, 1.0), 3.0, Float64(Float64(0.5 * t_0) * Float64(3.0 * cos(y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -8e-6], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 - N[(N[(N[Cos[y], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * 3.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999964e-6

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 57.1

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

   11. Applied rewrites 57.1%

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

   if -7.99999999999999964e-6 < x < 3.90000000000000005e-18

   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. 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 99.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sqrt.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sqrt.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if 3.90000000000000005e-18 < 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. 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 y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 65.0

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

   7. Applied rewrites 65.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \left(\cos y - \cos x\right) \cdot \left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, t\_1 \cdot 3\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_0, t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
   (if (<= x -8e-6)
     (/
      (fma
       (fma (sin x) -0.0625 (sin y))
       (* (* (sqrt 2.0) (sin x)) (- (cos x) 1.0))
       2.0)
      (fma (* t_0 (cos y)) 1.5 (* t_1 3.0)))
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_0 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (*
         (fma
          (fma -0.0625 (cos x) 0.0625)
          (* (pow (sin x) 2.0) (sqrt 2.0))
          2.0)
         0.3333333333333333)
        (fma (* 0.5 (cos y)) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double tmp;
	if (x <= -8e-6) {
		tmp = fma(fma(sin(x), -0.0625, sin(y)), ((sqrt(2.0) * sin(x)) * (cos(x) - 1.0)), 2.0) / fma((t_0 * cos(y)), 1.5, (t_1 * 3.0));
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = (fma(fma(-0.0625, cos(x), 0.0625), (pow(sin(x), 2.0) * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma((0.5 * cos(y)), t_0, t_1);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	tmp = 0.0
	if (x <= -8e-6)
		tmp = Float64(fma(fma(sin(x), -0.0625, sin(y)), Float64(Float64(sqrt(2.0) * sin(x)) * Float64(cos(x) - 1.0)), 2.0) / fma(Float64(t_0 * cos(y)), 1.5, Float64(t_1 * 3.0)));
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(Float64(fma(fma(-0.0625, cos(x), 0.0625), Float64((sin(x) ^ 2.0) * sqrt(2.0)), 2.0) * 0.3333333333333333) / fma(Float64(0.5 * cos(y)), t_0, t_1));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -8e-6], N[(N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(t$95$1 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$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[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999964e-6

   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-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. associate-/r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 57.1

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

   11. Applied rewrites 57.1%

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

   if -7.99999999999999964e-6 < x < 3.90000000000000005e-18

   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. 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 99.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sqrt.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   if 3.90000000000000005e-18 < 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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 64.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 64.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;y \leq -200:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_0 \cdot \cos y, 1.5, \mathsf{fma}\left(t\_2, \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00046:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, t\_2, 1\right), 3, \left(0.5 \cdot t\_0\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \end{array} \]

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))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= y -200.0)
     (/ t_1 (fma (* t_0 (cos y)) 1.5 (* (fma t_2 (cos x) 1.0) 3.0)))
     (if (<= y 0.00046)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (/ 6.0 (+ (sqrt 5.0) 3.0))))
       (/
        t_1
        (fma (fma (cos x) t_2 1.0) 3.0 (* (* 0.5 t_0) (* 3.0 (cos y)))))))))

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);
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (y <= -200.0) {
		tmp = t_1 / fma((t_0 * cos(y)), 1.5, (fma(t_2, cos(x), 1.0) * 3.0));
	} else if (y <= 0.00046) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (6.0 / (sqrt(5.0) + 3.0)));
	} else {
		tmp = t_1 / fma(fma(cos(x), t_2, 1.0), 3.0, ((0.5 * t_0) * (3.0 * cos(y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0)
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (y <= -200.0)
		tmp = Float64(t_1 / fma(Float64(t_0 * cos(y)), 1.5, Float64(fma(t_2, cos(x), 1.0) * 3.0)));
	elseif (y <= 0.00046)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(6.0 / Float64(sqrt(5.0) + 3.0))));
	else
		tmp = Float64(t_1 / fma(fma(cos(x), t_2, 1.0), 3.0, Float64(Float64(0.5 * t_0) * Float64(3.0 * cos(y)))));
	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[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$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[y, -200.0], N[(t$95$1 / N[(N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00046], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(6.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision] * 3.0 + N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(3.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -200

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right)\right), 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      14. lower-*.f64 99.2

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \cdot 3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\cos y \cdot \left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3\right)} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3\right) \cdot \cos y} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(\color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot 3\right) \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot 3\right) \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot 3\right)\right)} \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{3}{2}}\right) \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{3}{2} \cdot \cos y\right)} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(3 - \sqrt{5}\right) \cdot \color{blue}{\left(\cos y \cdot \frac{3}{2}\right)} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{3}{2}} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)}} \]

      13. lower-*.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}, 1.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       11. lower-sqrt.f64 65.3

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 65.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if -200 < y < 4.6000000000000001e-4

   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. 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 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 55.2

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 55.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 97.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 97.6%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)} \]

   if 4.6000000000000001e-4 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. 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 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   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. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. lower-sqrt.f64 57.1

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 57.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -200:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00046:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \left(3 \cdot \cos y\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{if}\;y \leq -200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.00046:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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)
          (fma
           (* (- 3.0 (sqrt 5.0)) (cos y))
           1.5
           (* (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0) 3.0)))))
   (if (<= y -200.0)
     t_0
     (if (<= y 0.00046)
       (/
        (fma (* (pow (sin x) 2.0) (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (fma
         (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
         3.0
         (/ 6.0 (+ (sqrt 5.0) 3.0))))
       t_0))))

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) / fma(((3.0 - sqrt(5.0)) * cos(y)), 1.5, (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0));
	double tmp;
	if (y <= -200.0) {
		tmp = t_0;
	} else if (y <= 0.00046) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (6.0 / (sqrt(5.0) + 3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)), 1.5, Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0) * 3.0)))
	tmp = 0.0
	if (y <= -200.0)
		tmp = t_0;
	elseif (y <= 0.00046)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(6.0 / Float64(sqrt(5.0) + 3.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -200.0], t$95$0, If[LessEqual[y, 0.00046], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(6.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -200 or 4.6000000000000001e-4 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. 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-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3 + \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3 - \sqrt{5}}{2} \cdot \cos y, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\cos y \cdot \frac{3 - \sqrt{5}}{2}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      10. clear-num N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\frac{1}{\frac{2}{3 - \sqrt{5}}}}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      11. associate-/r/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right)\right), 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, 3, \left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3\right)} \]

      14. lower-*.f64 99.2

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, \color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3}\right)} \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sin y - \frac{\sin x}{16}, \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}}{\mathsf{fma}\left(\cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot 3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \cdot 3 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\cos y \cdot \left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3\right)} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3\right) \cdot \cos y} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(\color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot 3\right) \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot 3\right) \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot 3\right)\right)} \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{3}{2}}\right) \cdot \cos y + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{3}{2} \cdot \cos y\right)} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\left(3 - \sqrt{5}\right) \cdot \color{blue}{\left(\cos y \cdot \frac{3}{2}\right)} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \cos y\right) \cdot \frac{3}{2}} + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right), \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), 1\right) \cdot 3\right)}} \]

      13. lower-*.f64 99.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\mathsf{fma}\left(\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}, 1.5, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right) \cdot 3\right)} \]

   8. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right), \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       6. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       10. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, \frac{3}{2}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right) \cdot 3\right)} \]

       11. lower-sqrt.f64 60.9

           \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   11. Applied rewrites 60.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]

   if -200 < y < 4.6000000000000001e-4

   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. 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 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 55.2

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 55.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 97.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 97.6%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -200:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \mathbf{elif}\;y \leq 0.00046:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right) \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := {\sin x}^{2} \cdot \sqrt{2}\\ t_2 := 3 - \sqrt{5}\\ t_3 := 0.5 \cdot \cos y\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(t\_0, \cos x, \mathsf{fma}\left(t\_2, t\_3, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), t\_1, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(t\_3, t\_2, \mathsf{fma}\left(t\_0, \cos x, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (* (pow (sin x) 2.0) (sqrt 2.0)))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* 0.5 (cos y))))
   (if (<= x -8e-6)
     (/
      (fma t_1 (fma (cos x) -0.0625 0.0625) 2.0)
      (* (fma t_0 (cos x) (fma t_2 t_3 1.0)) 3.0))
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_2 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (* (fma (fma -0.0625 (cos x) 0.0625) t_1 2.0) 0.3333333333333333)
        (fma t_3 t_2 (fma t_0 (cos x) 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = pow(sin(x), 2.0) * sqrt(2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 0.5 * cos(y);
	double tmp;
	if (x <= -8e-6) {
		tmp = fma(t_1, fma(cos(x), -0.0625, 0.0625), 2.0) / (fma(t_0, cos(x), fma(t_2, t_3, 1.0)) * 3.0);
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_2, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = (fma(fma(-0.0625, cos(x), 0.0625), t_1, 2.0) * 0.3333333333333333) / fma(t_3, t_2, fma(t_0, cos(x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64((sin(x) ^ 2.0) * sqrt(2.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(0.5 * cos(y))
	tmp = 0.0
	if (x <= -8e-6)
		tmp = Float64(fma(t_1, fma(cos(x), -0.0625, 0.0625), 2.0) / Float64(fma(t_0, cos(x), fma(t_2, t_3, 1.0)) * 3.0));
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_2, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(Float64(fma(fma(-0.0625, cos(x), 0.0625), t_1, 2.0) * 0.3333333333333333) / fma(t_3, t_2, fma(t_0, cos(x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-6], N[(N[(t$95$1 * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(t$95$2 * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + 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] * t$95$1 + 2.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(t$95$3 * t$95$2 + N[(t$95$0 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999964e-6

   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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

      6. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)}} \]

   7. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2} \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)}\right)\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{1}{2}} \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      11. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1}\right)} \]

   9. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, 1\right)}\right)} \]

   if -7.99999999999999964e-6 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   if 3.90000000000000005e-18 < 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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 64.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 64.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(t\_1, 0.5 \cdot \cos y, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (pow (sin x) 2.0) (sqrt 2.0))
          (fma (cos x) -0.0625 0.0625)
          2.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -8e-6)
     (/
      t_0
      (*
       (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) (fma t_1 (* 0.5 (cos y)) 1.0))
       3.0))
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_1 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        t_0
        (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (* t_1 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -8e-6) {
		tmp = t_0 / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), fma(t_1, (0.5 * cos(y)), 1.0)) * 3.0);
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_1, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = t_0 / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (t_1 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -8e-6)
		tmp = Float64(t_0 / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), fma(t_1, Float64(0.5 * cos(y)), 1.0)) * 3.0));
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_1, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(t_0 / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_1 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-6], N[(t$95$0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999964e-6

   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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

      6. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)}} \]

   7. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2} \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)}\right)\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{1}{2}} \cdot \left(\left(3 - \sqrt{5}\right) \cdot \cos y\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

      11. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1}\right)} \]

   9. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, 1\right)}\right)} \]

   if -7.99999999999999964e-6 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   if 3.90000000000000005e-18 < 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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 64.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 64.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 64.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2} \cdot \sqrt{2}\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_1, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), t\_0, 2\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_1 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (pow (sin x) 2.0) (sqrt 2.0))) (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -8e-6)
     (*
      (/
       0.3333333333333333
       (fma (* 0.5 (cos y)) t_1 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
      (fma (fma -0.0625 (cos x) 0.0625) t_0 2.0))
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_1 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (fma t_0 (fma (cos x) -0.0625 0.0625) 2.0)
        (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (* t_1 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0) * sqrt(2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -8e-6) {
		tmp = (0.3333333333333333 / fma((0.5 * cos(y)), t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma(fma(-0.0625, cos(x), 0.0625), t_0, 2.0);
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_1, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = fma(t_0, fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (t_1 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64((sin(x) ^ 2.0) * sqrt(2.0))
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -8e-6)
		tmp = Float64(Float64(0.3333333333333333 / fma(Float64(0.5 * cos(y)), t_1, fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0))) * fma(fma(-0.0625, cos(x), 0.0625), t_0, 2.0));
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_1, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(fma(t_0, fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_1 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-6], N[(N[(0.3333333333333333 / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999964e-6

   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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right)} \]

   if -7.99999999999999964e-6 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   if 3.90000000000000005e-18 < 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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 64.8

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 64.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 64.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), {\sin x}^{2} \cdot \sqrt{2}, 2\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma (cos x) -0.0625 0.0625)
           2.0)
          (fma 1.5 (fma (- (sqrt 5.0) 1.0) (cos x) (* t_0 (cos y))) 3.0))))
   (if (<= x -8e-6)
     t_1
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_0 (fma 1.5 (sqrt 5.0) 1.5)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma((sqrt(5.0) - 1.0), cos(x), (t_0 * cos(y))), 3.0);
	double tmp;
	if (x <= -8e-6) {
		tmp = t_1;
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(cos(x), -0.0625, 0.0625), 2.0) / fma(1.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), Float64(t_0 * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -8e-6)
		tmp = t_1;
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-6], t$95$1, If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999964e-6 or 3.90000000000000005e-18 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 60.3

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 60.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   if -7.99999999999999964e-6 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ t_1 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(t\_0, 3, \frac{6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_2, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(t\_0, 3, 1.5 \cdot t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
        (t_1 (fma -0.0625 (cos x) 0.0625))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -2.5e-5)
     (/
      (fma (* (pow (sin x) 2.0) (sqrt 2.0)) t_1 2.0)
      (fma t_0 3.0 (/ 6.0 (+ (sqrt 5.0) 3.0))))
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_2 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (fma (* (- 0.5 (* (cos (* 2.0 x)) 0.5)) (sqrt 2.0)) t_1 2.0)
        (fma t_0 3.0 (* 1.5 t_2)))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double t_1 = fma(-0.0625, cos(x), 0.0625);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -2.5e-5) {
		tmp = fma((pow(sin(x), 2.0) * sqrt(2.0)), t_1, 2.0) / fma(t_0, 3.0, (6.0 / (sqrt(5.0) + 3.0)));
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_2, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = fma(((0.5 - (cos((2.0 * x)) * 0.5)) * sqrt(2.0)), t_1, 2.0) / fma(t_0, 3.0, (1.5 * t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	t_1 = fma(-0.0625, cos(x), 0.0625)
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), t_1, 2.0) / fma(t_0, 3.0, Float64(6.0 / Float64(sqrt(5.0) + 3.0))));
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_2, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * x)) * 0.5)) * sqrt(2.0)), t_1, 2.0) / fma(t_0, 3.0, Float64(1.5 * t_2)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-5], N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(6.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(1.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e-5

   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 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 21.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 21.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 55.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.6%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)} \]

   if -2.50000000000000012e-5 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   if 3.90000000000000005e-18 < 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. 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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 22.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 22.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 63.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5}, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, \frac{6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{t\_0 - -0.5 \cdot t\_1} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(t\_0, 3, 1.5 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -2.5e-5)
     (*
      (/
       (fma (* (pow (sin x) 2.0) -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
       (- t_0 (* -0.5 t_1)))
      0.3333333333333333)
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_1 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (fma
         (* (- 0.5 (* (cos (* 2.0 x)) 0.5)) (sqrt 2.0))
         (fma -0.0625 (cos x) 0.0625)
         2.0)
        (fma t_0 3.0 (* 1.5 t_1)))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -2.5e-5) {
		tmp = (fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / (t_0 - (-0.5 * t_1))) * 0.3333333333333333;
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_1, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = fma(((0.5 - (cos((2.0 * x)) * 0.5)) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(t_0, 3.0, (1.5 * t_1));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / Float64(t_0 - Float64(-0.5 * t_1))) * 0.3333333333333333);
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_1, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * x)) * 0.5)) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(t_0, 3.0, Float64(1.5 * t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-5], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 - N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e-5

   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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

      6. associate--l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)}} \]

   7. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.8%

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]

   11. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) - \frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) - \frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3}} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) - \frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)} \cdot \frac{1}{3}} \]

   13. Applied rewrites 55.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin x}^{2}, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) - \left(3 - \sqrt{5}\right) \cdot -0.5} \cdot 0.3333333333333333} \]

   if -2.50000000000000012e-5 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   if 3.90000000000000005e-18 < 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. 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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 22.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 22.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 63.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5}, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) - -0.5 \cdot \left(3 - \sqrt{5}\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (fma -0.0625 (cos x) 0.0625)))
   (if (<= x -2.5e-5)
     (*
      (/
       (fma (* (pow (sin x) 2.0) (sqrt 2.0)) t_1 2.0)
       (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 1.0))
      0.3333333333333333)
     (if (<= x 3.9e-18)
       (/
        (+
         (* (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)))
         2.0)
        (fma (* 1.5 (cos y)) t_0 (fma 1.5 (sqrt 5.0) 1.5)))
       (/
        (fma (* (- 0.5 (* (cos (* 2.0 x)) 0.5)) (sqrt 2.0)) t_1 2.0)
        (fma (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0 (* 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(-0.0625, cos(x), 0.0625);
	double tmp;
	if (x <= -2.5e-5) {
		tmp = (fma((pow(sin(x), 2.0) * sqrt(2.0)), t_1, 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), t_0), 1.0)) * 0.3333333333333333;
	} else if (x <= 3.9e-18) {
		tmp = (((pow(sin(y), 2.0) * -0.0625) * ((1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma((1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5));
	} else {
		tmp = fma(((0.5 - (cos((2.0 * x)) * 0.5)) * sqrt(2.0)), t_1, 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.5 * t_0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(-0.0625, cos(x), 0.0625)
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), t_1, 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 1.0)) * 0.3333333333333333);
	elseif (x <= 3.9e-18)
		tmp = Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * sqrt(2.0))) + 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(1.5, sqrt(5.0), 1.5)));
	else
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * x)) * 0.5)) * sqrt(2.0)), t_1, 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.5 * t_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[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[x, -2.5e-5], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.9e-18], N[(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]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(1.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e-5

   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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 55.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -2.50000000000000012e-5 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)\right)} \]

      10. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   10. Applied rewrites 99.6%

       \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)} \]

   if 3.90000000000000005e-18 < 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. 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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 22.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 22.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 63.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5}, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left({\sin y}^{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) + 2}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, t\_0\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos y, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (fma -0.0625 (cos x) 0.0625)))
   (if (<= x -2.5e-5)
     (*
      (/
       (fma (* (pow (sin x) 2.0) (sqrt 2.0)) t_1 2.0)
       (fma 0.5 (fma (- (sqrt 5.0) 1.0) (cos x) t_0) 1.0))
      0.3333333333333333)
     (if (<= x 3.9e-18)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma t_0 (cos y) (sqrt 5.0)) 1.5))
       (/
        (fma (* (- 0.5 (* (cos (* 2.0 x)) 0.5)) (sqrt 2.0)) t_1 2.0)
        (fma (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0 (* 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(-0.0625, cos(x), 0.0625);
	double tmp;
	if (x <= -2.5e-5) {
		tmp = (fma((pow(sin(x), 2.0) * sqrt(2.0)), t_1, 2.0) / fma(0.5, fma((sqrt(5.0) - 1.0), cos(x), t_0), 1.0)) * 0.3333333333333333;
	} else if (x <= 3.9e-18) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(t_0, cos(y), sqrt(5.0)), 1.5);
	} else {
		tmp = fma(((0.5 - (cos((2.0 * x)) * 0.5)) * sqrt(2.0)), t_1, 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.5 * t_0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(-0.0625, cos(x), 0.0625)
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = Float64(Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), t_1, 2.0) / fma(0.5, fma(Float64(sqrt(5.0) - 1.0), cos(x), t_0), 1.0)) * 0.3333333333333333);
	elseif (x <= 3.9e-18)
		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(t_0, cos(y), sqrt(5.0)), 1.5));
	else
		tmp = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * x)) * 0.5)) * sqrt(2.0)), t_1, 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.5 * t_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[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]}, If[LessEqual[x, -2.5e-5], N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(0.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 3.9e-18], 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[(t$95$0 * N[Cos[y], $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e-5

   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

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower-cos.f64 56.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   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 55.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333} \]

   if -2.50000000000000012e-5 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}} \]

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 1.5\right)}} \]

   if 3.90000000000000005e-18 < 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. 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

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 22.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 22.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 63.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.7%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5}, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot t\_0\right)}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos y, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (- 0.5 (* (cos (* 2.0 x)) 0.5)) (sqrt 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (fma (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0) 3.0 (* 1.5 t_0)))))
   (if (<= x -2.5e-5)
     t_1
     (if (<= x 3.9e-18)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma t_0 (cos y) (sqrt 5.0)) 1.5))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(((0.5 - (cos((2.0 * x)) * 0.5)) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.5 * t_0));
	double tmp;
	if (x <= -2.5e-5) {
		tmp = t_1;
	} else if (x <= 3.9e-18) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(t_0, cos(y), sqrt(5.0)), 1.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * x)) * 0.5)) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.5 * t_0)))
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = t_1;
	elseif (x <= 3.9e-18)
		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(t_0, cos(y), sqrt(5.0)), 1.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-5], t$95$1, If[LessEqual[x, 3.9e-18], 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[(t$95$0 * N[Cos[y], $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000012e-5 or 3.90000000000000005e-18 < x

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 22.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 22.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   10. Applied rewrites 59.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.5%

       \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5}, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

   if -2.50000000000000012e-5 < x < 3.90000000000000005e-18

   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. 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 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      4. lower-cos.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      7. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}} \]

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 1.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 60.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (- 0.5 (* (cos (* 2.0 x)) 0.5)) (sqrt 2.0))
   (fma -0.0625 (cos x) 0.0625)
   2.0)
  (fma
   (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
   3.0
   (* 1.5 (- 3.0 (sqrt 5.0))))))

C

double code(double x, double y) {
	return fma(((0.5 - (cos((2.0 * x)) * 0.5)) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.5 * (3.0 - sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(fma(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * x)) * 0.5)) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.5 * Float64(3.0 - sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $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. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   4. lower-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

   8. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

   13. lower-sqrt.f64 57.5

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

7. Applied rewrites 57.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

10. Applied rewrites 57.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 57.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right)\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.5}, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]

13. Final simplification 57.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(2 \cdot x\right) \cdot 0.5\right) \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
14. Add Preprocessing

Alternative 25: 46.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \frac{-2}{\sqrt{5} + 3} \cdot \cos y\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   (fma
    (fma (sqrt 5.0) 0.5 -0.5)
    (cos x)
    (- 1.0 (* (/ -2.0 (+ (sqrt 5.0) 3.0)) (cos y))))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (1.0 - ((-2.0 / (sqrt(5.0) + 3.0)) * cos(y)))) * 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(1.0 - Float64(Float64(-2.0 / Float64(sqrt(5.0) + 3.0)) * cos(y)))) * 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(1.0 - N[(N[(-2.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 60.2

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 60.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   6. associate--l+ N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)}} \]

7. Applied rewrites 60.2%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 42.5%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \color{blue}{\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

    2. lift--.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \left(\frac{-1}{2} \cdot \color{blue}{\left(3 - \sqrt{5}\right)}\right) \cdot \cos y\right)} \]

    3. flip-- N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \left(\frac{-1}{2} \cdot \color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}\right) \cdot \cos y\right)} \]

    4. associate-*r/ N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \color{blue}{\frac{\frac{-1}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

    5. lift-sqrt.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \frac{\frac{-1}{2} \cdot \left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right)}{3 + \sqrt{5}} \cdot \cos y\right)} \]

    6. lift-sqrt.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \frac{\frac{-1}{2} \cdot \left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right)}{3 + \sqrt{5}} \cdot \cos y\right)} \]

    7. rem-square-sqrt N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \frac{\frac{-1}{2} \cdot \left(3 \cdot 3 - \color{blue}{5}\right)}{3 + \sqrt{5}} \cdot \cos y\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \frac{\frac{-1}{2} \cdot \left(\color{blue}{9} - 5\right)}{3 + \sqrt{5}} \cdot \cos y\right)} \]

    9. metadata-eval N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \frac{\frac{-1}{2} \cdot \color{blue}{4}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

    10. metadata-eval N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \frac{\color{blue}{-2}}{3 + \sqrt{5}} \cdot \cos y\right)} \]

    11. lower-/.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \color{blue}{\frac{-2}{3 + \sqrt{5}}} \cdot \cos y\right)} \]

    12. +-commutative N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \frac{-2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

    13. lower-+.f64 42.5

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \frac{-2}{\color{blue}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

12. Applied rewrites 42.5%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \color{blue}{\frac{-2}{\sqrt{5} + 3}} \cdot \cos y\right)} \]

13. Final simplification 42.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \frac{-2}{\sqrt{5} + 3} \cdot \cos y\right) \cdot 3} \]
14. Add Preprocessing

Alternative 26: 46.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1\right)\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   (fma
    (fma (sqrt 5.0) 0.5 -0.5)
    (cos x)
    (fma (* (- 3.0 (sqrt 5.0)) (cos y)) 0.5 1.0))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), fma(((3.0 - sqrt(5.0)) * cos(y)), 0.5, 1.0)) * 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), fma(Float64(Float64(3.0 - sqrt(5.0)) * cos(y)), 0.5, 1.0)) * 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] * 0.5 + 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

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 60.2

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 60.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   6. associate--l+ N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)}} \]

7. Applied rewrites 60.2%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 42.5%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]
11. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)} \]

    2. sub-neg N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{1 + \left(\mathsf{neg}\left(\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)\right)}\right)} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y}\right)\right)\right)} \]

    4. distribute-lft-neg-in N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) \cdot \cos y}\right)} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)}\right)\right) \cdot \cos y\right)} \]

    6. distribute-lft-neg-in N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(3 - \sqrt{5}\right)\right)} \cdot \cos y\right)} \]

    7. metadata-eval N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\color{blue}{\frac{1}{2}} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]

    8. *-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot \cos y\right)} \]

    9. metadata-eval N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos y\right)} \]

    10. div-inv N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

    11. lift-/.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y\right)} \]

    12. lift-*.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

    13. +-commutative N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y + 1}\right)} \]

12. Applied rewrites 42.5%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\mathsf{fma}\left(\cos y \cdot \left(3 - \sqrt{5}\right), 0.5, 1\right)}\right)} \]

13. Final simplification 42.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot \cos y, 0.5, 1\right)\right) \cdot 3} \]
14. Add Preprocessing

Alternative 27: 43.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)
   3.0
   (* 1.5 (- 3.0 (sqrt 5.0))))))

C

double code(double x, double y) {
	return 2.0 / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, (1.5 * (3.0 - sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(2.0 / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0), 3.0, Float64(1.5 * Float64(3.0 - sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $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. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   4. lower-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

   8. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

   13. lower-sqrt.f64 57.5

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

7. Applied rewrites 57.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

10. Applied rewrites 57.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, 1\right)}, 3, \frac{3}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \]
12. Step-by-step derivation

13. Applied rewrites 40.2%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)}, 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \]
14. Add Preprocessing

Alternative 28: 43.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (* (fma 0.5 (fma (- 3.0 (sqrt 5.0)) (cos y) (sqrt 5.0)) 0.5) 3.0)))

C

double code(double x, double y) {
	return 2.0 / (fma(0.5, fma((3.0 - sqrt(5.0)), cos(y), sqrt(5.0)), 0.5) * 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(fma(0.5, fma(Float64(3.0 - sqrt(5.0)), cos(y), sqrt(5.0)), 0.5) * 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\cos x \cdot \frac{-1}{16} + \left(\mathsf{neg}\left(1\right)\right) \cdot \frac{-1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{-1} \cdot \frac{-1}{16}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \cos x \cdot \frac{-1}{16} + \color{blue}{\frac{1}{16}}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \color{blue}{\mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right)}, 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   16. lower-cos.f64 60.2

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\color{blue}{\cos x}, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 60.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)}} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + 1\right)} - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)} \]

   6. associate--l+ N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \left(1 - \left(\mathsf{neg}\left(\frac{3 - \sqrt{5}}{2}\right)\right) \cdot \cos y\right)\right)}} \]

7. Applied rewrites 60.2%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 - \left(\frac{-1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]
9. Step-by-step derivation

10. Applied rewrites 42.5%

    \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 - \left(-0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y\right)} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) - \frac{-1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
12. Step-by-step derivation

    1. cancel-sub-sign-inv N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{2}{3 \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + \color{blue}{\frac{1}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

    3. associate-+r+ N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]

    4. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \frac{1}{2}\right)}} \]

    5. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \sqrt{5}\right)} + \frac{1}{2}\right)} \]

    6. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \left(\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right)} + \frac{1}{2}\right)} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}, \frac{1}{2}\right)}} \]

    8. *-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y} + \sqrt{5}, \frac{1}{2}\right)} \]

    9. lower-fma.f64 N/A

       \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right)}, \frac{1}{2}\right)} \]

    10. lower--.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \cos y, \sqrt{5}\right), \frac{1}{2}\right)} \]

    11. lower-sqrt.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \cos y, \sqrt{5}\right), \frac{1}{2}\right)} \]

    12. lower-cos.f64 N/A

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(3 - \sqrt{5}, \color{blue}{\cos y}, \sqrt{5}\right), \frac{1}{2}\right)} \]

    13. lower-sqrt.f64 39.3

        \[\leadsto \frac{2}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \color{blue}{\sqrt{5}}\right), 0.5\right)} \]

13. Applied rewrites 39.3%

    \[\leadsto \frac{2}{3 \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right)}} \]

14. Final simplification 39.3%

    \[\leadsto \frac{2}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(3 - \sqrt{5}, \cos y, \sqrt{5}\right), 0.5\right) \cdot 3} \]
15. Add Preprocessing

Alternative 29: 41.2% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma (fma 0.5 (sqrt 5.0) 0.5) 3.0 (* 1.5 (- 3.0 (sqrt 5.0))))))

C

double code(double x, double y) {
	return 2.0 / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, (1.5 * (3.0 - sqrt(5.0))));
}

Julia

function code(x, y)
	return Float64(2.0 / fma(fma(0.5, sqrt(5.0), 0.5), 3.0, Float64(1.5 * Float64(3.0 - sqrt(5.0)))))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision] * 3.0 + N[(1.5 * N[(3.0 - N[Sqrt[5.0], $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. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   4. lower-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2}\right)}\right)} \]

   8. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{3 \cdot \left(\frac{1}{2} \cdot \sqrt{5}\right) + 3 \cdot \frac{1}{2}}\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \sqrt{5}} + 3 \cdot \frac{1}{2}\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\frac{3}{2}} \cdot \sqrt{5} + 3 \cdot \frac{1}{2}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \frac{3}{2} \cdot \sqrt{5} + \color{blue}{\frac{3}{2}}\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{3}{2}, \sqrt{5}, \frac{3}{2}\right)}\right)} \]

   13. lower-sqrt.f64 57.5

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \color{blue}{\sqrt{5}}, 1.5\right)\right)} \]

7. Applied rewrites 57.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(1.5, \sqrt{5}, 1.5\right)\right)}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(1 + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

10. Applied rewrites 57.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\color{blue}{\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]
12. Step-by-step derivation

13. Applied rewrites 37.3%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right), 3, 1.5 \cdot \left(3 - \sqrt{5}\right)\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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))))))