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

Percentage Accurate: 99.3% → 99.4%

Time: 26.7s

Alternatives: 38

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \cos x - \sqrt{2} \cdot \cos y\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(\cos y \cdot 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
    (- (* (sqrt 2.0) (cos x)) (* (sqrt 2.0) (cos y)))))
  (+
   3.0
   (*
    3.0
    (fma
     (fma (sqrt 5.0) 0.5 -0.5)
     (cos x)
     (* (/ 4.0 (+ 3.0 (sqrt 5.0))) (* (cos y) 0.5)))))))

C

double code(double x, double y) {
	return (2.0 + ((fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * ((sqrt(2.0) * cos(x)) - (sqrt(2.0) * cos(y))))) / (3.0 + (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), ((4.0 / (3.0 + sqrt(5.0))) * (cos(y) * 0.5)))));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * Float64(Float64(sqrt(2.0) * cos(x)) - Float64(sqrt(2.0) * cos(y))))) / Float64(3.0 + Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) * Float64(cos(y) * 0.5))))))
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $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

4. Applied rewrites 99.4%

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

   1. lift-sqrt.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-+.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{\color{blue}{4}}{3 + \sqrt{5}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

6. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\frac{4}{3 + \sqrt{5}}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

8. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 99.5

      \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \cos x + \sqrt{2} \cdot \color{blue}{\left(-\cos y\right)}\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

10. Applied rewrites 99.5%

    \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \cos x + \sqrt{2} \cdot \left(-\cos y\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

11. Final simplification 99.5%

    \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \cos x - \sqrt{2} \cdot \cos y\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(\cos y \cdot 0.5\right)\right)} \]
12. Add Preprocessing

Alternative 2: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)\\ \mathbf{if}\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{t\_1} \leq 0.54:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ t_0 2.0))))))
   (if (<=
        (/
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (*
            (- (sin y) (/ (sin x) 16.0))
            (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))))
         t_1)
        0.54)
     (/
      (fma
       (- 1.0 (cos y))
       (* (- 0.5 (* 0.5 (cos (+ y y)))) (* -0.0625 (sqrt 2.0)))
       2.0)
      (+ 3.0 (* 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) -0.5))))
     (/ 2.0 t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0)));
	double tmp;
	if (((2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / t_1) <= 0.54) {
		tmp = fma((1.0 - cos(y)), ((0.5 - (0.5 * cos((y + y)))) * (-0.0625 * sqrt(2.0))), 2.0) / (3.0 + (3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5)));
	} else {
		tmp = 2.0 / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0))))
	tmp = 0.0
	if (Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))))) / t_1) <= 0.54)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(-0.0625 * sqrt(2.0))), 2.0) / Float64(3.0 + Float64(3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5))));
	else
		tmp = Float64(2.0 / 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[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 0.54], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y))))) < 0.54000000000000004

   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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 75.1

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

   10. Applied rewrites 75.1%

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

       1. lift-sin.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 N/A

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

       5. lift-cos.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift-*.f64 N/A

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

   12. Applied rewrites 75.1%

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

   if 0.54000000000000004 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y)))))

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. metadata-eval 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \leq 0.54:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)\\ \mathbf{if}\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{t\_1} \leq 0.54:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), -0.0625 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ t_0 2.0))))))
   (if (<=
        (/
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (*
            (- (sin y) (/ (sin x) 16.0))
            (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))))
         t_1)
        0.54)
     (/
      (fma
       (- 0.5 (* 0.5 (cos (+ y y))))
       (* -0.0625 (* (sqrt 2.0) (- 1.0 (cos y))))
       2.0)
      (* 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) 0.5)))
     (/ 2.0 t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0)));
	double tmp;
	if (((2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / t_1) <= 0.54) {
		tmp = fma((0.5 - (0.5 * cos((y + y)))), (-0.0625 * (sqrt(2.0) * (1.0 - cos(y)))), 2.0) / (3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5));
	} else {
		tmp = 2.0 / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0))))
	tmp = 0.0
	if (Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))))) / t_1) <= 0.54)
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(-0.0625 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5)));
	else
		tmp = Float64(2.0 / 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[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 0.54], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y))))) < 0.54000000000000004

   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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 75.1

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

   10. Applied rewrites 75.1%

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

   12. Applied rewrites 75.1%

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

   if 0.54000000000000004 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (sqrt.f64 #s(literal 2 binary64)) (-.f64 (sin.f64 x) (/.f64 (sin.f64 y) #s(literal 16 binary64)))) (-.f64 (sin.f64 y) (/.f64 (sin.f64 x) #s(literal 16 binary64)))) (-.f64 (cos.f64 x) (cos.f64 y)))) (*.f64 #s(literal 3 binary64) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (-.f64 (sqrt.f64 #s(literal 5 binary64)) #s(literal 1 binary64)) #s(literal 2 binary64)) (cos.f64 x))) (*.f64 (/.f64 (-.f64 #s(literal 3 binary64) (sqrt.f64 #s(literal 5 binary64))) #s(literal 2 binary64)) (cos.f64 y)))))

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. metadata-eval 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 25.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \leq 0.54:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), -0.0625 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(\cos y \cdot 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
    (* (sqrt 2.0) (- (cos x) (cos y)))))
  (+
   3.0
   (*
    3.0
    (fma
     (fma (sqrt 5.0) 0.5 -0.5)
     (cos x)
     (* (/ 4.0 (+ 3.0 (sqrt 5.0))) (* (cos y) 0.5)))))))

C

double code(double x, double y) {
	return (2.0 + ((fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * (sqrt(2.0) * (cos(x) - cos(y))))) / (3.0 + (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), ((4.0 / (3.0 + sqrt(5.0))) * (cos(y) * 0.5)))));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * Float64(sqrt(2.0) * Float64(cos(x) - cos(y))))) / Float64(3.0 + Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) * Float64(cos(y) * 0.5))))))
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $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

4. Applied rewrites 99.4%

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

   1. lift-sqrt.f64 N/A

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

   2. flip-- N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-+.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval 99.4

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{\color{blue}{4}}{3 + \sqrt{5}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

6. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{\frac{4}{3 + \sqrt{5}}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

8. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \color{blue}{\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

9. Final simplification 99.5%

   \[\leadsto \frac{2 + \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{4}{3 + \sqrt{5}} \cdot \left(\cos y \cdot 0.5\right)\right)} \]
10. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   0.3333333333333333
   (*
    (sqrt 2.0)
    (*
     (fma -0.0625 (sin y) (sin x))
     (* (- (cos x) (cos y)) (fma -0.0625 (sin x) (sin y)))))
   0.6666666666666666)
  (fma
   (cos y)
   (* 0.5 (- 3.0 (sqrt 5.0)))
   (fma (cos x) (fma (sqrt 5.0) 0.5 -0.5) 1.0))))

C

double code(double x, double y) {
	return fma(0.3333333333333333, (sqrt(2.0) * (fma(-0.0625, sin(y), sin(x)) * ((cos(x) - cos(y)) * fma(-0.0625, sin(x), sin(y))))), 0.6666666666666666) / fma(cos(y), (0.5 * (3.0 - sqrt(5.0))), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0));
}

Julia

function code(x, y)
	return Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64(fma(-0.0625, sin(y), sin(x)) * Float64(Float64(cos(x) - cos(y)) * fma(-0.0625, sin(x), sin(y))))), 0.6666666666666666) / fma(cos(y), Float64(0.5 * Float64(3.0 - sqrt(5.0))), fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), 1.0)))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.4%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-sqrt.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 8: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x - \cos y\\ t_3 := 2 + t\_2 \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot t\_0\right)\\ t_4 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0021:\\ \;\;\;\;\frac{t\_3}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_4}{2}\right) + \cos y \cdot \frac{t\_1}{2}\right)}\\ \mathbf{elif}\;x \leq 0.225:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), -1.5\right) + \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, t\_4 \cdot \mathsf{fma}\left(x \cdot x, -0.0020833333333333333, 0.0625\right), \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (+ 2.0 (* t_2 (* (* (sin x) (sqrt 2.0)) t_0))))
        (t_4 (+ (sqrt 5.0) -1.0)))
   (if (<= x -0.0021)
     (/
      t_3
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_4 2.0))) (* (cos y) (/ t_1 2.0)))))
     (if (<= x 0.225)
       (/
        (+ 2.0 (* t_2 (* t_0 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))))
        (+
         (fma 1.5 (fma (cos y) t_1 (sqrt 5.0)) -1.5)
         (fma
          (* x x)
          (fma
           (* x x)
           (* t_4 (fma (* x x) -0.0020833333333333333 0.0625))
           (fma (sqrt 5.0) -0.75 0.75))
          3.0)))
       (/
        t_3
        (+
         3.0
         (*
          3.0
          (fma
           (fma (sqrt 5.0) 0.5 -0.5)
           (cos x)
           (* (* (cos y) 0.5) t_1)))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = 2.0 + (t_2 * ((sin(x) * sqrt(2.0)) * t_0));
	double t_4 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -0.0021) {
		tmp = t_3 / (3.0 * ((1.0 + (cos(x) * (t_4 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	} else if (x <= 0.225) {
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / (fma(1.5, fma(cos(y), t_1, sqrt(5.0)), -1.5) + fma((x * x), fma((x * x), (t_4 * fma((x * x), -0.0020833333333333333, 0.0625)), fma(sqrt(5.0), -0.75, 0.75)), 3.0));
	} else {
		tmp = t_3 / (3.0 + (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), ((cos(y) * 0.5) * t_1))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(2.0 + Float64(t_2 * Float64(Float64(sin(x) * sqrt(2.0)) * t_0)))
	t_4 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -0.0021)
		tmp = Float64(t_3 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_4 / 2.0))) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	elseif (x <= 0.225)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))))) / Float64(fma(1.5, fma(cos(y), t_1, sqrt(5.0)), -1.5) + fma(Float64(x * x), fma(Float64(x * x), Float64(t_4 * fma(Float64(x * x), -0.0020833333333333333, 0.0625)), fma(sqrt(5.0), -0.75, 0.75)), 3.0)));
	else
		tmp = Float64(t_3 / Float64(3.0 + Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(cos(y) * 0.5) * t_1)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(t$95$2 * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.0021], N[(t$95$3 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$4 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.225], N[(N[(2.0 + N[(t$95$2 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(t$95$4 * N[(N[(x * x), $MachinePrecision] * -0.0020833333333333333 + 0.0625), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * -0.75 + 0.75), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(3.0 + N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00209999999999999987

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if -0.00209999999999999987 < x < 0.225000000000000006

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + {x}^{2} \cdot \left(\frac{-3}{4} \cdot \left(\sqrt{5} - 1\right) + {x}^{2} \cdot \left(\frac{-1}{480} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{16} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right) + \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.0020833333333333333, 0.0625\right), \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right)\right), 3\right)}} \]

   if 0.225000000000000006 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 71.6

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

   7. Applied rewrites 71.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0021:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.225:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -1.5\right) + \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.0020833333333333333, 0.0625\right), \mathsf{fma}\left(\sqrt{5}, -0.75, 0.75\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ \mathbf{if}\;x \leq -0.42:\\ \;\;\;\;\frac{t\_2}{t\_1}\\ \mathbf{elif}\;x \leq 0.225:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ t_0 2.0)))))
        (t_2
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))))))
   (if (<= x -0.42)
     (/ t_2 t_1)
     (if (<= x 0.225)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (*
           (fma (sin y) -0.0625 (sin x))
           (*
            (fma (sin x) -0.0625 (sin y))
            (fma
             (* x x)
             (fma
              (* x x)
              (fma (* x x) -0.001388888888888889 0.041666666666666664)
              -0.5)
             (- 1.0 (cos y)))))))
        t_1)
       (/
        t_2
        (+
         3.0
         (*
          3.0
          (fma
           (fma (sqrt 5.0) 0.5 -0.5)
           (cos x)
           (* (* (cos y) 0.5) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0)));
	double t_2 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double tmp;
	if (x <= -0.42) {
		tmp = t_2 / t_1;
	} else if (x <= 0.225) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (fma(sin(x), -0.0625, sin(y)) * fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), (1.0 - cos(y))))))) / t_1;
	} else {
		tmp = t_2 / (3.0 + (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), ((cos(y) * 0.5) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0))))
	t_2 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	tmp = 0.0
	if (x <= -0.42)
		tmp = Float64(t_2 / t_1);
	elseif (x <= 0.225)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), Float64(1.0 - cos(y))))))) / t_1);
	else
		tmp = Float64(t_2 / Float64(3.0 + Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(cos(y) * 0.5) * 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[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.42], N[(t$95$2 / t$95$1), $MachinePrecision], If[LessEqual[x, 0.225], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$2 / N[(3.0 + N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.419999999999999984

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -0.419999999999999984 < x < 0.225000000000000006

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right)}\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \left(1 - \cos y\right)\right)}\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1 - \cos y\right)}\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1 - \cos y\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), \color{blue}{1 - \cos y}\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      17. lower-cos.f64 99.7

          \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \color{blue}{\cos y}\right)\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)}\right)\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.225000000000000006 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 71.6

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

   7. Applied rewrites 71.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.42:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.225:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.42:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.225:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right) \cdot \left(--3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 + 3 \cdot \mathsf{fma}\left(t\_2, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -0.42)
     (/
      t_1
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ t_0 2.0)))))
     (if (<= x 0.225)
       (/
        (fma
         (sqrt 2.0)
         (*
          (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
          (-
           (fma
            (* x x)
            (fma
             x
             (* x (fma (* x x) -0.001388888888888889 0.041666666666666664))
             -0.5)
            1.0)
           (cos y)))
         2.0)
        (* (fma (cos x) t_2 (fma (cos y) (* 0.5 t_0) 1.0)) (- -3.0)))
       (/ t_1 (+ 3.0 (* 3.0 (fma t_2 (cos x) (* (* (cos y) 0.5) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -0.42) {
		tmp = t_1 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (x <= 0.225) {
		tmp = fma(sqrt(2.0), ((fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * (fma((x * x), fma(x, (x * fma((x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0) - cos(y))), 2.0) / (fma(cos(x), t_2, fma(cos(y), (0.5 * t_0), 1.0)) * -(-3.0));
	} else {
		tmp = t_1 / (3.0 + (3.0 * fma(t_2, cos(x), ((cos(y) * 0.5) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -0.42)
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (x <= 0.225)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0) - cos(y))), 2.0) / Float64(fma(cos(x), t_2, fma(cos(y), Float64(0.5 * t_0), 1.0)) * Float64(-(-3.0))));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(3.0 * fma(t_2, cos(x), Float64(Float64(cos(y) * 0.5) * 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[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.42], N[(t$95$1 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.225], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * (--3.0)), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(3.0 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.419999999999999984

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -0.419999999999999984 < x < 0.225000000000000006

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right)} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) - \cos y\right)} \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      6. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right) + \color{blue}{\frac{-1}{2}}, 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right), \frac{-1}{2}\right)}, 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      13. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      15. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) - \cos y\right) \cdot \left(\mathsf{fma}\left(\sin y, \frac{-1}{16}, \sin x\right) \cdot \mathsf{fma}\left(\sin x, \frac{-1}{16}, \sin y\right)\right), 2\right)\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

      17. lower-cos.f64 99.7

          \[\leadsto \frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \color{blue}{\cos y}\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

   9. Applied rewrites 99.7%

      \[\leadsto \frac{-\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \cos y\right)} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

   if 0.225000000000000006 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 71.6

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

   7. Applied rewrites 71.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.42:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.225:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot \left(--3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.28:\\ \;\;\;\;\frac{t\_1}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.118:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{3 + 3 \cdot \mathsf{fma}\left(t\_2, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0))))))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= x -0.28)
     (/
      t_1
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ t_0 2.0)))))
     (if (<= x 0.118)
       (/
        (-
         (fma
          (sqrt 2.0)
          (*
           (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
           (-
            (fma (* x x) (fma x (* x 0.041666666666666664) -0.5) 1.0)
            (cos y)))
          2.0))
        (* (fma (cos x) t_2 (fma (cos y) (* 0.5 t_0) 1.0)) -3.0))
       (/ t_1 (+ 3.0 (* 3.0 (fma t_2 (cos x) (* (* (cos y) 0.5) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (x <= -0.28) {
		tmp = t_1 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (x <= 0.118) {
		tmp = -fma(sqrt(2.0), ((fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * (fma((x * x), fma(x, (x * 0.041666666666666664), -0.5), 1.0) - cos(y))), 2.0) / (fma(cos(x), t_2, fma(cos(y), (0.5 * t_0), 1.0)) * -3.0);
	} else {
		tmp = t_1 / (3.0 + (3.0 * fma(t_2, cos(x), ((cos(y) * 0.5) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (x <= -0.28)
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (x <= 0.118)
		tmp = Float64(Float64(-fma(sqrt(2.0), Float64(Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.041666666666666664), -0.5), 1.0) - cos(y))), 2.0)) / Float64(fma(cos(x), t_2, fma(cos(y), Float64(0.5 * t_0), 1.0)) * -3.0));
	else
		tmp = Float64(t_1 / Float64(3.0 + Float64(3.0 * fma(t_2, cos(x), Float64(Float64(cos(y) * 0.5) * 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[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.28], N[(t$95$1 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.118], N[((-N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]) / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 + N[(3.0 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.28000000000000003

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -0.28000000000000003 < x < 0.11799999999999999

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 99.6

          \[\leadsto \frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) - \color{blue}{\cos y}\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{-\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) - \cos y\right)} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3} \]

   if 0.11799999999999999 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 71.6

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

   7. Applied rewrites 71.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.28:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.118:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), 1\right) - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{t\_2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{elif}\;x \leq 0.052:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, t\_1, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{3 + 3 \cdot \mathsf{fma}\left(t\_1, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))))))
   (if (<= x -0.03)
     (/
      t_2
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ t_0 2.0)))))
     (if (<= x 0.052)
       (/
        (-
         (fma
          (sqrt 2.0)
          (*
           (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
           (fma x (* x -0.5) (- 1.0 (cos y))))
          2.0))
        (* (fma (cos x) t_1 (fma (cos y) (* 0.5 t_0) 1.0)) -3.0))
       (/ t_2 (+ 3.0 (* 3.0 (fma t_1 (cos x) (* (* (cos y) 0.5) t_0)))))))))

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 = 2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))));
	double tmp;
	if (x <= -0.03) {
		tmp = t_2 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (x <= 0.052) {
		tmp = -fma(sqrt(2.0), ((fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * fma(x, (x * -0.5), (1.0 - cos(y)))), 2.0) / (fma(cos(x), t_1, fma(cos(y), (0.5 * t_0), 1.0)) * -3.0);
	} else {
		tmp = t_2 / (3.0 + (3.0 * fma(t_1, cos(x), ((cos(y) * 0.5) * t_0))));
	}
	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(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	tmp = 0.0
	if (x <= -0.03)
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (x <= 0.052)
		tmp = Float64(Float64(-fma(sqrt(2.0), Float64(Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * fma(x, Float64(x * -0.5), Float64(1.0 - cos(y)))), 2.0)) / Float64(fma(cos(x), t_1, fma(cos(y), Float64(0.5 * t_0), 1.0)) * -3.0));
	else
		tmp = Float64(t_2 / Float64(3.0 + Float64(3.0 * fma(t_1, cos(x), Float64(Float64(cos(y) * 0.5) * 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[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.03], N[(t$95$2 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.052], N[((-N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]) / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 + N[(3.0 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.029999999999999999

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -0.029999999999999999 < x < 0.0519999999999999976

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 0.0519999999999999976 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 71.6

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

   7. Applied rewrites 71.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.052:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_1, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.052:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, t\_1, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))))
          (+ 3.0 (* 3.0 (fma t_1 (cos x) (* (* (cos y) 0.5) t_0)))))))
   (if (<= x -0.03)
     t_2
     (if (<= x 0.052)
       (/
        (-
         (fma
          (sqrt 2.0)
          (*
           (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
           (fma x (* x -0.5) (- 1.0 (cos y))))
          2.0))
        (* (fma (cos x) t_1 (fma (cos y) (* 0.5 t_0) 1.0)) -3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = (2.0 + ((cos(x) - cos(y)) * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 + (3.0 * fma(t_1, cos(x), ((cos(y) * 0.5) * t_0))));
	double tmp;
	if (x <= -0.03) {
		tmp = t_2;
	} else if (x <= 0.052) {
		tmp = -fma(sqrt(2.0), ((fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * fma(x, (x * -0.5), (1.0 - cos(y)))), 2.0) / (fma(cos(x), t_1, fma(cos(y), (0.5 * t_0), 1.0)) * -3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 + Float64(3.0 * fma(t_1, cos(x), Float64(Float64(cos(y) * 0.5) * t_0)))))
	tmp = 0.0
	if (x <= -0.03)
		tmp = t_2;
	elseif (x <= 0.052)
		tmp = Float64(Float64(-fma(sqrt(2.0), Float64(Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * fma(x, Float64(x * -0.5), Float64(1.0 - cos(y)))), 2.0)) / Float64(fma(cos(x), t_1, fma(cos(y), Float64(0.5 * t_0), 1.0)) * -3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.03], t$95$2, If[LessEqual[x, 0.052], N[((-N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]) / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.029999999999999999 or 0.0519999999999999976 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if -0.029999999999999999 < x < 0.0519999999999999976

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 99.4

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

   9. Applied rewrites 99.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.03:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.052:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\\ t_2 := \frac{2 + t\_0 \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, t\_1\right)}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot t\_0\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\sqrt{5} + -1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* (* (cos y) 0.5) (- 3.0 (sqrt 5.0))))
        (t_2
         (/
          (+
           2.0
           (* t_0 (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))))
          (+ 3.0 (* 3.0 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) t_1))))))
   (if (<= x -2.15e-7)
     t_2
     (if (<= x 0.0255)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (*
           (fma (sin y) -0.0625 (sin x))
           (* (fma (sin x) -0.0625 (sin y)) t_0))))
        (* 3.0 (+ 1.0 (fma (+ (sqrt 5.0) -1.0) (fma -0.25 (* x x) 0.5) t_1))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (cos(y) * 0.5) * (3.0 - sqrt(5.0));
	double t_2 = (2.0 + (t_0 * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 + (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), t_1)));
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_2;
	} else if (x <= 0.0255) {
		tmp = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (fma(sin(x), -0.0625, sin(y)) * t_0)))) / (3.0 * (1.0 + fma((sqrt(5.0) + -1.0), fma(-0.25, (x * x), 0.5), t_1)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(cos(y) * 0.5) * Float64(3.0 - sqrt(5.0)))
	t_2 = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 + Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), t_1))))
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = t_2;
	elseif (x <= 0.0255)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(fma(sin(x), -0.0625, sin(y)) * t_0)))) / Float64(3.0 * Float64(1.0 + fma(Float64(sqrt(5.0) + -1.0), fma(-0.25, Float64(x * x), 0.5), t_1))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(t$95$0 * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], t$95$2, If[LessEqual[x, 0.0255], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-0.25 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e-7 or 0.0254999999999999984 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 66.3

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

   7. Applied rewrites 66.3%

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

   if -2.1500000000000001e-7 < x < 0.0254999999999999984

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. associate-+r+ N/A

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

   7. Applied rewrites 99.4%

      \[\leadsto \frac{\sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right)\right) + 2}{3 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(\sqrt{5} + -1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(0.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0255:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\sqrt{5} + -1, \mathsf{fma}\left(-0.25, x \cdot x, 0.5\right), \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \frac{2 + t\_0 \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_1, \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_0, \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, t\_1, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2
         (/
          (+
           2.0
           (* t_0 (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))))
          (+
           3.0
           (* 3.0 (fma t_1 (cos x) (* (* (cos y) 0.5) (- 3.0 (sqrt 5.0)))))))))
   (if (<= x -0.024)
     t_2
     (if (<= x 0.034)
       (/
        (fma
         (* (sqrt 2.0) t_0)
         (fma
          -0.0625
          (- 0.5 (* 0.5 (cos (+ y y))))
          (* x (fma (sin y) 1.00390625 (* -0.0625 x))))
         2.0)
        (fma 3.0 (fma (cos x) t_1 (* (cos y) (- 1.5 (* (sqrt 5.0) 0.5)))) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = (2.0 + (t_0 * ((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 + (3.0 * fma(t_1, cos(x), ((cos(y) * 0.5) * (3.0 - sqrt(5.0))))));
	double tmp;
	if (x <= -0.024) {
		tmp = t_2;
	} else if (x <= 0.034) {
		tmp = fma((sqrt(2.0) * t_0), fma(-0.0625, (0.5 - (0.5 * cos((y + y)))), (x * fma(sin(y), 1.00390625, (-0.0625 * x)))), 2.0) / fma(3.0, fma(cos(x), t_1, (cos(y) * (1.5 - (sqrt(5.0) * 0.5)))), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 + Float64(3.0 * fma(t_1, cos(x), Float64(Float64(cos(y) * 0.5) * Float64(3.0 - sqrt(5.0)))))))
	tmp = 0.0
	if (x <= -0.024)
		tmp = t_2;
	elseif (x <= 0.034)
		tmp = Float64(fma(Float64(sqrt(2.0) * t_0), fma(-0.0625, Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(x * fma(sin(y), 1.00390625, Float64(-0.0625 * x)))), 2.0) / fma(3.0, fma(cos(x), t_1, Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(t$95$0 * N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.024], t$95$2, If[LessEqual[x, 0.034], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625 + N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.024 or 0.034000000000000002 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if -0.024 < x < 0.034000000000000002

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 99.3

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

   9. Applied rewrites 99.3%

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

   11. Applied rewrites 99.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \mathbf{elif}\;x \leq 0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right) \cdot \left(--3\right)}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (sqrt 2.0)
           (*
            (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
            (- 1.0 (cos y)))
           2.0)
          (*
           (fma
            (cos x)
            (fma (sqrt 5.0) 0.5 -0.5)
            (fma (cos y) (* 0.5 t_0) 1.0))
           (- -3.0)))))
   (if (<= y -1.32e-5)
     t_1
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(sqrt(2.0), ((fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * (1.0 - cos(y))), 2.0) / (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(cos(y), (0.5 * t_0), 1.0)) * -(-3.0));
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_1;
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(sqrt(2.0), Float64(Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))) * Float64(1.0 - cos(y))), 2.0) / Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(cos(y), Float64(0.5 * t_0), 1.0)) * Float64(-(-3.0))))
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = t_1;
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * (--3.0)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], t$95$1, If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000007e-5 or 2e-30 < 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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 58.7

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

   9. Applied rewrites 58.7%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot \left(--3\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right) \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot \left(--3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\cos x, t\_1, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right)} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, t\_1, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \end{array} \end{array} \]

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 -0.024)
     (*
      (/ 1.0 (fma (cos x) t_1 (fma (cos y) (* 0.5 t_0) 1.0)))
      (*
       0.3333333333333333
       (fma
        (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
        (+ 0.5 (* -0.5 (cos (+ x x))))
        2.0)))
     (if (<= x 0.034)
       (/
        (fma
         (* (sqrt 2.0) (- (cos x) (cos y)))
         (fma
          -0.0625
          (- 0.5 (* 0.5 (cos (+ y y))))
          (* x (fma (sin y) 1.00390625 (* -0.0625 x))))
         2.0)
        (fma 3.0 (fma (cos x) t_1 (* (cos y) (- 1.5 (* (sqrt 5.0) 0.5)))) 3.0))
       (/
        (+
         2.0
         (*
          (* (* (sin x) (sqrt 2.0)) (- (sin y) (/ (sin x) 16.0)))
          (+ (cos x) -1.0)))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ t_0 2.0)))))))))

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 <= -0.024) {
		tmp = (1.0 / fma(cos(x), t_1, fma(cos(y), (0.5 * t_0), 1.0))) * (0.3333333333333333 * fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0));
	} else if (x <= 0.034) {
		tmp = fma((sqrt(2.0) * (cos(x) - cos(y))), fma(-0.0625, (0.5 - (0.5 * cos((y + y)))), (x * fma(sin(y), 1.00390625, (-0.0625 * x)))), 2.0) / fma(3.0, fma(cos(x), t_1, (cos(y) * (1.5 - (sqrt(5.0) * 0.5)))), 3.0);
	} else {
		tmp = (2.0 + (((sin(x) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	}
	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 <= -0.024)
		tmp = Float64(Float64(1.0 / fma(cos(x), t_1, fma(cos(y), Float64(0.5 * t_0), 1.0))) * Float64(0.3333333333333333 * fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0)));
	elseif (x <= 0.034)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), fma(-0.0625, Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(x * fma(sin(y), 1.00390625, Float64(-0.0625 * x)))), 2.0) / fma(3.0, fma(cos(x), t_1, Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))), 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.024], N[(N[(1.0 / N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.034], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625 + N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.024

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.3%

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

   if -0.024 < x < 0.034000000000000002

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 99.3

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

   9. Applied rewrites 99.3%

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

   11. Applied rewrites 99.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}} \]

   if 0.034000000000000002 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. metadata-eval 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 69.1

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

   8. Applied rewrites 69.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin x \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right)} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_2, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

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)))
   (if (<= x -0.024)
     (*
      (/ 1.0 (fma (cos x) t_2 (fma (cos y) (* 0.5 t_0) 1.0)))
      (*
       0.3333333333333333
       (fma
        (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
        (+ 0.5 (* -0.5 (cos (+ x x))))
        2.0)))
     (if (<= x 0.034)
       (/
        (fma
         (* (sqrt 2.0) t_1)
         (fma
          -0.0625
          (- 0.5 (* 0.5 (cos (+ y y))))
          (* x (fma (sin y) 1.00390625 (* -0.0625 x))))
         2.0)
        (fma 3.0 (fma (cos x) t_2 (* (cos y) (- 1.5 (* (sqrt 5.0) 0.5)))) 3.0))
       (/
        (+ 2.0 (* t_1 (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (+ 3.0 (* 3.0 (fma t_2 (cos x) (* (* (cos y) 0.5) t_0)))))))))

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 tmp;
	if (x <= -0.024) {
		tmp = (1.0 / fma(cos(x), t_2, fma(cos(y), (0.5 * t_0), 1.0))) * (0.3333333333333333 * fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0));
	} else if (x <= 0.034) {
		tmp = fma((sqrt(2.0) * t_1), fma(-0.0625, (0.5 - (0.5 * cos((y + y)))), (x * fma(sin(y), 1.00390625, (-0.0625 * x)))), 2.0) / fma(3.0, fma(cos(x), t_2, (cos(y) * (1.5 - (sqrt(5.0) * 0.5)))), 3.0);
	} else {
		tmp = (2.0 + (t_1 * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 + (3.0 * fma(t_2, cos(x), ((cos(y) * 0.5) * t_0))));
	}
	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)
	tmp = 0.0
	if (x <= -0.024)
		tmp = Float64(Float64(1.0 / fma(cos(x), t_2, fma(cos(y), Float64(0.5 * t_0), 1.0))) * Float64(0.3333333333333333 * fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0)));
	elseif (x <= 0.034)
		tmp = Float64(fma(Float64(sqrt(2.0) * t_1), fma(-0.0625, Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(x * fma(sin(y), 1.00390625, Float64(-0.0625 * x)))), 2.0) / fma(3.0, fma(cos(x), t_2, Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))), 3.0));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 + Float64(3.0 * fma(t_2, cos(x), Float64(Float64(cos(y) * 0.5) * 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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -0.024], N[(N[(1.0 / N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.034], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625 + N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.024

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.3%

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

   if -0.024 < x < 0.034000000000000002

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 99.3

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

   9. Applied rewrites 99.3%

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

   11. Applied rewrites 99.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}} \]

   if 0.034000000000000002 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 69.0

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

   7. Applied rewrites 69.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(-0.0625, 0.5 - 0.5 \cdot \cos \left(y + y\right), x \cdot \mathsf{fma}\left(\sin y, 1.00390625, -0.0625 \cdot x\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, t\_1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (/
          (+
           2.0
           (*
            (sqrt 2.0)
            (* (fma (sin y) -0.0625 (sin x)) (* (sin y) (- 1.0 (cos y))))))
          (*
           3.0
           (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) (* (cos y) (/ t_0 2.0)))))))
   (if (<= y -1.32e-5)
     t_2
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) t_1 t_0) 1.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = (2.0 + (sqrt(2.0) * (fma(sin(y), -0.0625, sin(x)) * (sin(y) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_2;
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), t_1, t_0), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))))
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = t_2;
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), t_1, t_0), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], t$95$2, If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000007e-5 or 2e-30 < 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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 58.4

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

   7. Applied rewrites 58.4%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \mathsf{fma}\left(\cos x, t\_1, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{t\_2} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(1.00390625, \sin y, -0.0625 \cdot x\right), -0.0625 \cdot {\sin y}^{2}\right), 2\right)}{t\_2 \cdot \left(--3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_1, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (fma (cos x) t_1 (fma (cos y) (* 0.5 t_0) 1.0))))
   (if (<= x -2.15e-7)
     (*
      (/ 1.0 t_2)
      (*
       0.3333333333333333
       (fma
        (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
        (+ 0.5 (* -0.5 (cos (+ x x))))
        2.0)))
     (if (<= x 0.016)
       (/
        (fma
         (sqrt 2.0)
         (*
          (- 1.0 (cos y))
          (fma
           x
           (fma 1.00390625 (sin y) (* -0.0625 x))
           (* -0.0625 (pow (sin y) 2.0))))
         2.0)
        (* t_2 (- -3.0)))
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (+ 3.0 (* 3.0 (fma t_1 (cos x) (* (* (cos y) 0.5) t_0)))))))))

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(cos(x), t_1, fma(cos(y), (0.5 * t_0), 1.0));
	double tmp;
	if (x <= -2.15e-7) {
		tmp = (1.0 / t_2) * (0.3333333333333333 * fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0));
	} else if (x <= 0.016) {
		tmp = fma(sqrt(2.0), ((1.0 - cos(y)) * fma(x, fma(1.00390625, sin(y), (-0.0625 * x)), (-0.0625 * pow(sin(y), 2.0)))), 2.0) / (t_2 * -(-3.0));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 + (3.0 * fma(t_1, cos(x), ((cos(y) * 0.5) * t_0))));
	}
	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(cos(x), t_1, fma(cos(y), Float64(0.5 * t_0), 1.0))
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = Float64(Float64(1.0 / t_2) * Float64(0.3333333333333333 * fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0)));
	elseif (x <= 0.016)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(1.0 - cos(y)) * fma(x, fma(1.00390625, sin(y), Float64(-0.0625 * x)), Float64(-0.0625 * (sin(y) ^ 2.0)))), 2.0) / Float64(t_2 * Float64(-(-3.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 + Float64(3.0 * fma(t_1, cos(x), Float64(Float64(cos(y) * 0.5) * 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[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.016], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(x * N[(1.00390625 * N[Sin[y], $MachinePrecision] + N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$2 * (--3.0)), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1500000000000001e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 99.3

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-cos.f64 99.1

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

   12. Applied rewrites 99.1%

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

   if 0.016 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 69.0

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

   7. Applied rewrites 69.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(1 - \cos y\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(1.00390625, \sin y, -0.0625 \cdot x\right), -0.0625 \cdot {\sin y}^{2}\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot \left(--3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\cos x, t\_2, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right)} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_1 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(1.00390625, \sin y, -0.0625 \cdot x\right), -0.0625 \cdot {\sin y}^{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 0.5\right) \cdot \left(--3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_2, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

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)))
   (if (<= x -2.15e-7)
     (*
      (/ 1.0 (fma (cos x) t_2 (fma (cos y) (* 0.5 t_0) 1.0)))
      (*
       0.3333333333333333
       (fma
        (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
        (+ 0.5 (* -0.5 (cos (+ x x))))
        2.0)))
     (if (<= x 0.016)
       (/
        (fma
         (sqrt 2.0)
         (*
          t_1
          (fma
           x
           (fma 1.00390625 (sin y) (* -0.0625 x))
           (* -0.0625 (pow (sin y) 2.0))))
         2.0)
        (* (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) 0.5) (- -3.0)))
       (/
        (+ 2.0 (* t_1 (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
        (+ 3.0 (* 3.0 (fma t_2 (cos x) (* (* (cos y) 0.5) t_0)))))))))

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 tmp;
	if (x <= -2.15e-7) {
		tmp = (1.0 / fma(cos(x), t_2, fma(cos(y), (0.5 * t_0), 1.0))) * (0.3333333333333333 * fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0));
	} else if (x <= 0.016) {
		tmp = fma(sqrt(2.0), (t_1 * fma(x, fma(1.00390625, sin(y), (-0.0625 * x)), (-0.0625 * pow(sin(y), 2.0)))), 2.0) / (fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5) * -(-3.0));
	} else {
		tmp = (2.0 + (t_1 * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 + (3.0 * fma(t_2, cos(x), ((cos(y) * 0.5) * t_0))));
	}
	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)
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = Float64(Float64(1.0 / fma(cos(x), t_2, fma(cos(y), Float64(0.5 * t_0), 1.0))) * Float64(0.3333333333333333 * fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0)));
	elseif (x <= 0.016)
		tmp = Float64(fma(sqrt(2.0), Float64(t_1 * fma(x, fma(1.00390625, sin(y), Float64(-0.0625 * x)), Float64(-0.0625 * (sin(y) ^ 2.0)))), 2.0) / Float64(fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5) * Float64(-(-3.0))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 + Float64(3.0 * fma(t_2, cos(x), Float64(Float64(cos(y) * 0.5) * 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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], N[(N[(1.0 / N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.016], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(x * N[(1.00390625 * N[Sin[y], $MachinePrecision] + N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * (--3.0)), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1500000000000001e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 99.3

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in x around 0

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

       1. distribute-lft-in N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-sqrt.f64 99.0

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

   12. Applied rewrites 99.0%

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

   if 0.016 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 69.0

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

   7. Applied rewrites 69.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)} \cdot \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\right)\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(1.00390625, \sin y, -0.0625 \cdot x\right), -0.0625 \cdot {\sin y}^{2}\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right) \cdot \left(--3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \cos x - \cos y\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, t\_2 \cdot \left(-0.0625 \cdot t\_0\right), 2\right)}{\mathsf{fma}\left(\cos x, t\_1, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_3, 1\right)\right) \cdot -3}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_3\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t\_0\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_1, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (- (cos x) (cos y)))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= y -1.32e-5)
     (/
      (- (fma (sqrt 2.0) (* t_2 (* -0.0625 t_0)) 2.0))
      (* (fma (cos x) t_1 (fma (cos y) (* 0.5 t_3) 1.0)) -3.0))
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_3) 1.0))
       (/
        (+ 2.0 (* t_2 (* -0.0625 (* (sqrt 2.0) t_0))))
        (+ 3.0 (* 3.0 (fma t_1 (cos x) (* (* (cos y) 0.5) t_3)))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = cos(x) - cos(y);
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -1.32e-5) {
		tmp = -fma(sqrt(2.0), (t_2 * (-0.0625 * t_0)), 2.0) / (fma(cos(x), t_1, fma(cos(y), (0.5 * t_3), 1.0)) * -3.0);
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_3), 1.0);
	} else {
		tmp = (2.0 + (t_2 * (-0.0625 * (sqrt(2.0) * t_0)))) / (3.0 + (3.0 * fma(t_1, cos(x), ((cos(y) * 0.5) * t_3))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = Float64(Float64(-fma(sqrt(2.0), Float64(t_2 * Float64(-0.0625 * t_0)), 2.0)) / Float64(fma(cos(x), t_1, fma(cos(y), Float64(0.5 * t_3), 1.0)) * -3.0));
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_3), 1.0));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(-0.0625 * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 + Float64(3.0 * fma(t_1, cos(x), Float64(Float64(cos(y) * 0.5) * t_3)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], N[((-N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(-0.0625 * t$95$0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]) / N[(N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32000000000000007e-5

   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

   4. Applied rewrites 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 62.0

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

   9. Applied rewrites 62.0%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

   if 2e-30 < 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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 54.8

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

   7. Applied rewrites 54.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right) \cdot -3}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{-\mathsf{fma}\left(\sqrt{2}, \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, 1\right)\right) \cdot -3}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (-
           (fma
            (sqrt 2.0)
            (* (- (cos x) (cos y)) (* -0.0625 (pow (sin y) 2.0)))
            2.0))
          (*
           (fma
            (cos x)
            (fma (sqrt 5.0) 0.5 -0.5)
            (fma (cos y) (* 0.5 t_0) 1.0))
           -3.0))))
   (if (<= y -1.32e-5)
     t_1
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = -fma(sqrt(2.0), ((cos(x) - cos(y)) * (-0.0625 * pow(sin(y), 2.0))), 2.0) / (fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(cos(y), (0.5 * t_0), 1.0)) * -3.0);
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_1;
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(-fma(sqrt(2.0), Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * (sin(y) ^ 2.0))), 2.0)) / Float64(fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(cos(y), Float64(0.5 * t_0), 1.0)) * -3.0))
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = t_1;
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]) / N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], t$95$1, If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000007e-5 or 2e-30 < 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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 57.9

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

   9. Applied rewrites 57.9%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{2 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* -0.0625 (pow (sin y) 2.0))
           (* (sqrt 2.0) (- 1.0 (cos y)))
           2.0)
          (+
           3.0
           (*
            3.0
            (fma
             (fma (sqrt 5.0) 0.5 -0.5)
             (cos x)
             (/ (* 2.0 (cos y)) (+ 3.0 (sqrt 5.0)))))))))
   (if (<= y -1.32e-5)
     t_0
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (- 3.0 (sqrt 5.0))) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / (3.0 + (3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), ((2.0 * cos(y)) / (3.0 + sqrt(5.0))))));
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_0;
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), (3.0 - sqrt(5.0))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 + Float64(3.0 * fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(2.0 * cos(y)) / Float64(3.0 + sqrt(5.0)))))))
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = t_0;
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(3.0 - sqrt(5.0))), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(2.0 * N[Cos[y], $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], t$95$0, If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000007e-5 or 2e-30 < 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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 57.8

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

   7. Applied rewrites 57.8%

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

      1. lift-sqrt.f64 N/A

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

      2. flip-- N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 57.9

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

   9. Applied rewrites 57.9%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{2 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \frac{2 \cdot \cos y}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{3 + 3 \cdot \mathsf{fma}\left(\cos y, 0.5 \cdot t\_1, \cos x \cdot t\_2\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{3 + 3 \cdot \mathsf{fma}\left(t\_2, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* -0.0625 (pow (sin y) 2.0))
          (* (sqrt 2.0) (- 1.0 (cos y)))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= y -1.32e-5)
     (/ t_0 (+ 3.0 (* 3.0 (fma (cos y) (* 0.5 t_1) (* (cos x) t_2)))))
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_1) 1.0))
       (/ t_0 (+ 3.0 (* 3.0 (fma t_2 (cos x) (* (* (cos y) 0.5) t_1)))))))))

C

double code(double x, double y) {
	double t_0 = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_0 / (3.0 + (3.0 * fma(cos(y), (0.5 * t_1), (cos(x) * t_2))));
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_1), 1.0);
	} else {
		tmp = t_0 / (3.0 + (3.0 * fma(t_2, cos(x), ((cos(y) * 0.5) * t_1))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = Float64(t_0 / Float64(3.0 + Float64(3.0 * fma(cos(y), Float64(0.5 * t_1), Float64(cos(x) * t_2)))));
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_1), 1.0));
	else
		tmp = Float64(t_0 / Float64(3.0 + Float64(3.0 * fma(t_2, cos(x), Float64(Float64(cos(y) * 0.5) * t_1)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], N[(t$95$0 / N[(3.0 + N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 + N[(3.0 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32000000000000007e-5

   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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 61.9

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 61.9%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

   if 2e-30 < 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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 54.6

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

   7. Applied rewrites 54.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{2} \cdot \left(1 - \cos y\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), -0.0625 \cdot t\_1, 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, \cos x \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, t\_1, 2\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_2, \cos x, \left(\cos y \cdot 0.5\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (* (sqrt 2.0) (- 1.0 (cos y))))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5)))
   (if (<= y -1.32e-5)
     (/
      (fma (- 0.5 (* 0.5 (cos (+ y y)))) (* -0.0625 t_1) 2.0)
      (fma 3.0 (fma (cos y) (* 0.5 t_0) (* (cos x) t_2)) 3.0))
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) t_1 2.0)
        (+ 3.0 (* 3.0 (fma t_2 (cos x) (* (* (cos y) 0.5) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(2.0) * (1.0 - cos(y));
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double tmp;
	if (y <= -1.32e-5) {
		tmp = fma((0.5 - (0.5 * cos((y + y)))), (-0.0625 * t_1), 2.0) / fma(3.0, fma(cos(y), (0.5 * t_0), (cos(x) * t_2)), 3.0);
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), t_1, 2.0) / (3.0 + (3.0 * fma(t_2, cos(x), ((cos(y) * 0.5) * t_0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(2.0) * Float64(1.0 - cos(y)))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(-0.0625 * t_1), 2.0) / fma(3.0, fma(cos(y), Float64(0.5 * t_0), Float64(cos(x) * t_2)), 3.0));
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), t_1, 2.0) / Float64(3.0 + Float64(3.0 * fma(t_2, cos(x), Float64(Float64(cos(y) * 0.5) * 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[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32000000000000007e-5

   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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 61.9

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 61.9%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

   if 2e-30 < 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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 54.6

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

   7. Applied rewrites 54.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), -0.0625 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(\cos y \cdot 0.5\right) \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 78.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), -0.0625 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_1, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{t\_2}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* 0.5 (cos (+ y y))))
          (* -0.0625 (* (sqrt 2.0) (- 1.0 (cos y))))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (fma
          3.0
          (fma (cos y) (* 0.5 t_1) (* (cos x) (fma (sqrt 5.0) 0.5 -0.5)))
          3.0)))
   (if (<= y -1.32e-5)
     (/ t_0 t_2)
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_1) 1.0))
       (/ 1.0 (/ t_2 t_0))))))

C

double code(double x, double y) {
	double t_0 = fma((0.5 - (0.5 * cos((y + y)))), (-0.0625 * (sqrt(2.0) * (1.0 - cos(y)))), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(3.0, fma(cos(y), (0.5 * t_1), (cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0);
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_0 / t_2;
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_1), 1.0);
	} else {
		tmp = 1.0 / (t_2 / t_0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(-0.0625 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(3.0, fma(cos(y), Float64(0.5 * t_1), Float64(cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0)
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = Float64(t_0 / t_2);
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_1), 1.0));
	else
		tmp = Float64(1.0 / Float64(t_2 / t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], N[(t$95$0 / t$95$2), $MachinePrecision], If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32000000000000007e-5

   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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 61.9

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 61.9%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

   if 2e-30 < 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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 54.6

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

   7. Applied rewrites 54.6%

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

   9. Applied rewrites 54.5%

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

Alternative 28: 78.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), -0.0625 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_0, \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right), 3\right)}\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (- 0.5 (* 0.5 (cos (+ y y))))
           (* -0.0625 (* (sqrt 2.0) (- 1.0 (cos y))))
           2.0)
          (fma
           3.0
           (fma (cos y) (* 0.5 t_0) (* (cos x) (fma (sqrt 5.0) 0.5 -0.5)))
           3.0))))
   (if (<= y -1.32e-5)
     t_1
     (if (<= y 2e-30)
       (/
        (fma
         0.3333333333333333
         (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((0.5 - (0.5 * cos((y + y)))), (-0.0625 * (sqrt(2.0) * (1.0 - cos(y)))), 2.0) / fma(3.0, fma(cos(y), (0.5 * t_0), (cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0);
	double tmp;
	if (y <= -1.32e-5) {
		tmp = t_1;
	} else if (y <= 2e-30) {
		tmp = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(-0.0625 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))), 2.0) / fma(3.0, fma(cos(y), Float64(0.5 * t_0), Float64(cos(x) * fma(sqrt(5.0), 0.5, -0.5))), 3.0))
	tmp = 0.0
	if (y <= -1.32e-5)
		tmp = t_1;
	elseif (y <= 2e-30)
		tmp = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$0), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.32e-5], t$95$1, If[LessEqual[y, 2e-30], N[(N[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32000000000000007e-5 or 2e-30 < 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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 57.8

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

   7. Applied rewrites 57.8%

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

   9. Applied rewrites 57.8%

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

   if -1.32000000000000007e-5 < y < 2e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 79.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := 0.5 \cdot t\_2\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, t\_1, \mathsf{fma}\left(\cos y, t\_3, 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_2, \cos y \cdot 0.5, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, t\_1, \cos y \cdot t\_3\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
          (+ 0.5 (* -0.5 (cos (+ x x))))
          2.0))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (* 0.5 t_2)))
   (if (<= x -2.15e-7)
     (* t_0 (/ 0.3333333333333333 (fma (cos x) t_1 (fma (cos y) t_3 1.0))))
     (if (<= x 0.016)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (+ 3.0 (* 3.0 (fma t_2 (* (cos y) 0.5) t_1))))
       (/ t_0 (fma 3.0 (fma (cos x) t_1 (* (cos y) t_3)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0);
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 0.5 * t_2;
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_0 * (0.3333333333333333 / fma(cos(x), t_1, fma(cos(y), t_3, 1.0)));
	} else if (x <= 0.016) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / (3.0 + (3.0 * fma(t_2, (cos(y) * 0.5), t_1)));
	} else {
		tmp = t_0 / fma(3.0, fma(cos(x), t_1, (cos(y) * t_3)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0)
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(0.5 * t_2)
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = Float64(t_0 * Float64(0.3333333333333333 / fma(cos(x), t_1, fma(cos(y), t_3, 1.0))));
	elseif (x <= 0.016)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 + Float64(3.0 * fma(t_2, Float64(cos(y) * 0.5), t_1))));
	else
		tmp = Float64(t_0 / fma(3.0, fma(cos(x), t_1, Float64(cos(y) * t_3)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$2), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], N[(t$95$0 * N[(0.3333333333333333 / N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.016], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$2 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1500000000000001e-7

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 59.8%

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

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 98.5

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

   10. Applied rewrites 98.5%

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

       1. lift-cos.f64 N/A

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

       2. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right) + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       3. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)} + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\sqrt{5}}\right) + \frac{-1}{2}\right) \cdot 3} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       7. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2} + \sqrt{5} \cdot \frac{1}{2}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \color{blue}{\frac{1}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       9. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       10. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       11. associate-*l/ N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

   12. Applied rewrites 98.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \cdot 3} \]

   if 0.016 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 68.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)}{\mathsf{fma}\left(3, \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 79.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)\\ t_3 := \mathsf{fma}\left(\cos x, t\_0, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_1, 1\right)\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_2 \cdot \frac{0.3333333333333333}{t\_3}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_1, \cos y \cdot 0.5, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot t\_2}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (fma
          (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
          (+ 0.5 (* -0.5 (cos (+ x x))))
          2.0))
        (t_3 (fma (cos x) t_0 (fma (cos y) (* 0.5 t_1) 1.0))))
   (if (<= x -2.15e-7)
     (* t_2 (/ 0.3333333333333333 t_3))
     (if (<= x 0.016)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (+ 3.0 (* 3.0 (fma t_1 (* (cos y) 0.5) t_0))))
       (/ (* 0.3333333333333333 t_2) t_3)))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0);
	double t_3 = fma(cos(x), t_0, fma(cos(y), (0.5 * t_1), 1.0));
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_2 * (0.3333333333333333 / t_3);
	} else if (x <= 0.016) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / (3.0 + (3.0 * fma(t_1, (cos(y) * 0.5), t_0)));
	} else {
		tmp = (0.3333333333333333 * t_2) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0)
	t_3 = fma(cos(x), t_0, fma(cos(y), Float64(0.5 * t_1), 1.0))
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = Float64(t_2 * Float64(0.3333333333333333 / t_3));
	elseif (x <= 0.016)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 + Float64(3.0 * fma(t_1, Float64(cos(y) * 0.5), t_0))));
	else
		tmp = Float64(Float64(0.3333333333333333 * t_2) / t_3);
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], N[(t$95$2 * N[(0.3333333333333333 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.016], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1500000000000001e-7

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 59.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}} \]

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      10. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)} \cdot 3} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot 3} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot 3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{-1}{2}}\right) \cdot 3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{-1}{2}\right)} \cdot 3} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}}, \frac{-1}{2}\right) \cdot 3} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)}, \frac{-1}{2}\right) \cdot 3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      10. lower-sqrt.f64 98.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right), -0.5\right) \cdot 3} \]

   10. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)} \cdot 3} \]
   11. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right) + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       3. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)} + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\sqrt{5}}\right) + \frac{-1}{2}\right) \cdot 3} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       7. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2} + \sqrt{5} \cdot \frac{1}{2}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \color{blue}{\frac{1}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       9. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       10. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       11. associate-*l/ N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

   12. Applied rewrites 98.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \cdot 3} \]

   if 0.016 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 68.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 69.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 79.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, t\_0, \mathsf{fma}\left(\cos y, 0.5 \cdot t\_1, 1\right)\right)}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_1, \cos y \cdot 0.5, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (*
          (fma
           (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
           (+ 0.5 (* -0.5 (cos (+ x x))))
           2.0)
          (/
           0.3333333333333333
           (fma (cos x) t_0 (fma (cos y) (* 0.5 t_1) 1.0))))))
   (if (<= x -2.15e-7)
     t_2
     (if (<= x 0.016)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (+ 3.0 (* 3.0 (fma t_1 (* (cos y) 0.5) t_0))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), (0.5 + (-0.5 * cos((x + x)))), 2.0) * (0.3333333333333333 / fma(cos(x), t_0, fma(cos(y), (0.5 * t_1), 1.0)));
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_2;
	} else if (x <= 0.016) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / (3.0 + (3.0 * fma(t_1, (cos(y) * 0.5), t_0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), 2.0) * Float64(0.3333333333333333 / fma(cos(x), t_0, fma(cos(y), Float64(0.5 * t_1), 1.0))))
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = t_2;
	elseif (x <= 0.016)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 + Float64(3.0 * fma(t_1, Float64(cos(y) * 0.5), t_0))));
	else
		tmp = t_2;
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], t$95$2, If[LessEqual[x, 0.016], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e-7 or 0.016 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 63.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}} \]

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      10. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)} \cdot 3} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot 3} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot 3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{-1}{2}}\right) \cdot 3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{-1}{2}\right)} \cdot 3} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}}, \frac{-1}{2}\right) \cdot 3} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)}, \frac{-1}{2}\right) \cdot 3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      10. lower-sqrt.f64 98.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right), -0.5\right) \cdot 3} \]

   10. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)} \cdot 3} \]
   11. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right) + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       3. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)} + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\sqrt{5}}\right) + \frac{-1}{2}\right) \cdot 3} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       7. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2} + \sqrt{5} \cdot \frac{1}{2}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \color{blue}{\frac{1}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       9. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       10. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       11. associate-*l/ N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

   12. Applied rewrites 98.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 0.5 + -0.5 \cdot \cos \left(x + x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 32: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(t\_0, \cos y \cdot 0.5, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           0.3333333333333333
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
           0.6666666666666666)
          (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))))
   (if (<= x -2.15e-7)
     t_1
     (if (<= x 0.016)
       (/
        (fma (* -0.0625 (pow (sin y) 2.0)) (* (sqrt 2.0) (- 1.0 (cos y))) 2.0)
        (+ 3.0 (* 3.0 (fma t_0 (* (cos y) 0.5) (fma (sqrt 5.0) 0.5 -0.5)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_1;
	} else if (x <= 0.016) {
		tmp = fma((-0.0625 * pow(sin(y), 2.0)), (sqrt(2.0) * (1.0 - cos(y))), 2.0) / (3.0 + (3.0 * fma(t_0, (cos(y) * 0.5), fma(sqrt(5.0), 0.5, -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0))
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = t_1;
	elseif (x <= 0.016)
		tmp = Float64(fma(Float64(-0.0625 * (sin(y) ^ 2.0)), Float64(sqrt(2.0) * Float64(1.0 - cos(y))), 2.0) / Float64(3.0 + Float64(3.0 * fma(t_0, Float64(cos(y) * 0.5), fma(sqrt(5.0), 0.5, -0.5)))));
	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[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], t$95$1, If[LessEqual[x, 0.016], N[(N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e-7 or 0.016 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 63.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      10. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)} \cdot 3} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot 3} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot 3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{-1}{2}}\right) \cdot 3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{-1}{2}\right)} \cdot 3} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}}, \frac{-1}{2}\right) \cdot 3} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)}, \frac{-1}{2}\right) \cdot 3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      10. lower-sqrt.f64 98.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right), -0.5\right) \cdot 3} \]

   10. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)} \cdot 3} \]
   11. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right) + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       3. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)} + \sqrt{5}\right) + \frac{-1}{2}\right) \cdot 3} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \color{blue}{\sqrt{5}}\right) + \frac{-1}{2}\right) \cdot 3} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       7. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \frac{1}{2} + \sqrt{5} \cdot \frac{1}{2}\right)} + \frac{-1}{2}\right) \cdot 3} \]

       8. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) \cdot \color{blue}{\frac{1}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       9. div-inv N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{\cos y \cdot \left(3 - \sqrt{5}\right)}{2}} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       10. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\frac{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \cos y}}{2} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       11. associate-*l/ N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2}} \cdot \cos y + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\left(\color{blue}{\frac{3 - \sqrt{5}}{2} \cdot \cos y} + \sqrt{5} \cdot \frac{1}{2}\right) + \frac{-1}{2}\right) \cdot 3} \]

   12. Applied rewrites 98.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, 0.5 \cdot \cos y, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)} \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 33: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           0.3333333333333333
           (* (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)))
           0.6666666666666666)
          (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))))
   (if (<= x -2.15e-7)
     t_1
     (if (<= x 0.016)
       (/
        (fma
         (- 1.0 (cos y))
         (* (- 0.5 (* 0.5 (cos (+ y y)))) (* -0.0625 (sqrt 2.0)))
         2.0)
        (+ 3.0 (* 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) -0.5))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(0.3333333333333333, (pow(sin(x), 2.0) * (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_1;
	} else if (x <= 0.016) {
		tmp = fma((1.0 - cos(y)), ((0.5 - (0.5 * cos((y + y)))) * (-0.0625 * sqrt(2.0))), 2.0) / (3.0 + (3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(0.3333333333333333, Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625))), 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0))
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = t_1;
	elseif (x <= 0.016)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(-0.0625 * sqrt(2.0))), 2.0) / Float64(3.0 + Float64(3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), -0.5))));
	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[(0.3333333333333333 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], t$95$1, If[LessEqual[x, 0.016], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e-7 or 0.016 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 63.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   8. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      10. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)} \cdot 3} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot 3} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot 3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{-1}{2}}\right) \cdot 3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{-1}{2}\right)} \cdot 3} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}}, \frac{-1}{2}\right) \cdot 3} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)}, \frac{-1}{2}\right) \cdot 3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      10. lower-sqrt.f64 98.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right), -0.5\right) \cdot 3} \]

   10. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)} \cdot 3} \]
   11. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       6. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

   12. Applied rewrites 98.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right) \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 34: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (sqrt 2.0) (pow (sin x) 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (fma 1.5 (- (fma (+ (sqrt 5.0) -1.0) (cos x) 3.0) (sqrt 5.0)) 3.0))))
   (if (<= x -2.15e-7)
     t_0
     (if (<= x 0.016)
       (/
        (fma
         (- 1.0 (cos y))
         (* (- 0.5 (* 0.5 (cos (+ y y)))) (* -0.0625 (sqrt 2.0)))
         2.0)
        (+
         3.0
         (* 3.0 (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -0.5))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(2.0) * pow(sin(x), 2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, (fma((sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 3.0);
	double tmp;
	if (x <= -2.15e-7) {
		tmp = t_0;
	} else if (x <= 0.016) {
		tmp = fma((1.0 - cos(y)), ((0.5 - (0.5 * cos((y + y)))) * (-0.0625 * sqrt(2.0))), 2.0) / (3.0 + (3.0 * fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(sqrt(2.0) * (sin(x) ^ 2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, Float64(fma(Float64(sqrt(5.0) + -1.0), cos(x), 3.0) - sqrt(5.0)), 3.0))
	tmp = 0.0
	if (x <= -2.15e-7)
		tmp = t_0;
	elseif (x <= 0.016)
		tmp = Float64(fma(Float64(1.0 - cos(y)), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))) * Float64(-0.0625 * sqrt(2.0))), 2.0) / Float64(3.0 + Float64(3.0 * fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-7], t$95$0, If[LessEqual[x, 0.016], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1500000000000001e-7 or 0.016 < x

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 62.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x - 1\right)\right)} \cdot \frac{-1}{16} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot {\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{{\sin x}^{2}}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\color{blue}{\sin x}}^{2}, \frac{-1}{16} \cdot \left(\cos x - 1\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      14. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

      17. lower-cos.f64 62.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \color{blue}{\cos x}, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)} \]

   if -2.1500000000000001e-7 < x < 0.016

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 99.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      9. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

      10. lower-cos.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)} \cdot 3} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot 3} \]

      2. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot 3} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \left(\frac{1}{2} \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{-1}{2}}\right) \cdot 3} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right), \frac{-1}{2}\right)} \cdot 3} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}}, \frac{-1}{2}\right) \cdot 3} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)}, \frac{-1}{2}\right) \cdot 3} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

      10. lower-sqrt.f64 98.5

          \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}}\right), -0.5\right) \cdot 3} \]

   10. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)} \cdot 3} \]
   11. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       2. lift-pow.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right)} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       5. lift-cos.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right)\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       6. lift--.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}\right) + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), \frac{-1}{2}\right) \cdot 3} \]

   12. Applied rewrites 98.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 + \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right) \cdot 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)}\\ \mathbf{elif}\;x \leq 0.016:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \left(0.5 - 0.5 \cdot \cos \left(y + y\right)\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{3 + 3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin x}^{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3\right) - \sqrt{5}, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 35: 45.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
    (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))

C

double code(double x, double y) {
	return 2.0 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
end function

Java

public static double code(double x, double y) {
	return 2.0 / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
}

Python

def code(x, y):
	return 2.0 / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))

Julia

function code(x, y)
	return Float64(2.0 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
end

Wolfram

code[x_, y_] := N[(2.0 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x - 1\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-cos.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\cos x} + \left(\mathsf{neg}\left(1\right)\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. metadata-eval 66.2

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + \color{blue}{-1}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 66.2%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\cos x + -1\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

8. Applied rewrites 47.9%

   \[\leadsto \frac{\color{blue}{2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Final simplification 47.9%

   \[\leadsto \frac{2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)} \]
10. Add Preprocessing

Alternative 36: 42.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(3 - \sqrt{5}, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, 3\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (- 3.0 (sqrt 5.0))
   1.5
   (fma (fma (sqrt 5.0) 0.5 -0.5) (* (cos x) 3.0) 3.0))))

C

double code(double x, double y) {
	return 2.0 / fma((3.0 - sqrt(5.0)), 1.5, fma(fma(sqrt(5.0), 0.5, -0.5), (cos(x) * 3.0), 3.0));
}

Julia

function code(x, y)
	return Float64(2.0 / fma(Float64(3.0 - sqrt(5.0)), 1.5, fma(fma(sqrt(5.0), 0.5, -0.5), Float64(cos(x) * 3.0), 3.0)))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 1.5 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * 3.0), $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

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(1 - \cos y\right)}, 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right) \cdot 3} \]

   10. lower-cos.f64 62.0

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \color{blue}{\cos y}\right), 2\right)}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

7. Applied rewrites 62.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right) \cdot 3} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{2}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3}} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3 + \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3\right)} + 3} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3 + \left(\left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3 + 3\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)} \cdot 3 + \left(\left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3 + 3\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{2}{\color{blue}{\left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot 3\right)} + \left(\left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3 + 3\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{2}{\left(3 - \sqrt{5}\right) \cdot \color{blue}{\frac{3}{2}} + \left(\left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3 + 3\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(3 - \sqrt{5}, \frac{3}{2}, \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3 + 3\right)}} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{3 - \sqrt{5}}, \frac{3}{2}, \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3 + 3\right)} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(3 - \color{blue}{\sqrt{5}}, \frac{3}{2}, \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3 + 3\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(3 - \sqrt{5}, \frac{3}{2}, \color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot \cos x\right)} \cdot 3 + 3\right)} \]

   12. associate-*l* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(3 - \sqrt{5}, \frac{3}{2}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + 3\right)} \]

10. Applied rewrites 46.2%

    \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(3 - \sqrt{5}, 1.5, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x \cdot 3, 3\right)\right)}} \]
11. Add Preprocessing

Alternative 37: 42.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, -1 + \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  0.6666666666666666
  (fma 0.5 (+ -1.0 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0))) 1.0)))

C

double code(double x, double y) {
	return 0.6666666666666666 / fma(0.5, (-1.0 + fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0))), 1.0);
}

Julia

function code(x, y)
	return Float64(0.6666666666666666 / fma(0.5, Float64(-1.0 + fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0))), 1.0))
end

Wolfram

code[x_, y_] := N[(0.6666666666666666 / N[(0.5 * N[(-1.0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 66.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right)\right), 1\right)} \]

   12. metadata-eval 45.1

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 1\right)} \]

8. Applied rewrites 45.1%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]
9. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos y} \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} + -1\right), 1\right)} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \color{blue}{\sqrt{5}}\right) + \left(\sqrt{5} + -1\right), 1\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \color{blue}{\left(3 - \sqrt{5}\right)} + \left(\sqrt{5} + -1\right), 1\right)} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\color{blue}{\sqrt{5}} + -1\right), 1\right)} \]

   5. associate-+r+ N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1}, 1\right)} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\cos y \cdot \left(3 - \sqrt{5}\right) + \sqrt{5}\right) + -1}, 1\right)} \]

   7. lower-fma.f64 45.1

      \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right)} + -1, 1\right)} \]

10. Applied rewrites 45.1%

    \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right) + -1}, 1\right)} \]

11. Final simplification 45.1%

    \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, -1 + \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 1\right)} \]
12. Add Preprocessing

Alternative 38: 40.2% accurate, 940.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 66.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 1} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 1\right)}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 1\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 1\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 1\right)} \]

   9. sub-neg N/A

      \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 1\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\frac{2}{3}}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right)\right), 1\right)} \]

   12. metadata-eval 45.1

       \[\leadsto \frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 1\right)} \]

8. Applied rewrites 45.1%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3}} \]
10. Step-by-step derivation

11. Applied rewrites 43.7%

    \[\leadsto \color{blue}{0.3333333333333333} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))))