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

Percentage Accurate: 99.3% → 99.4%

Time: 19.0s

Alternatives: 24

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

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (* (sqrt 2.0) (- (cos x) (cos y)))
   (* (fma (sin y) -0.0625 (sin x)) (fma (sin x) -0.0625 (sin y)))
   2.0)
  (fma
   1.5
   (fma (cos x) (- (sqrt 5.0) 1.0) (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
   3.0)))

C

double code(double x, double y) {
	return fma((sqrt(2.0) * (cos(x) - cos(y))), (fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(fma(sin(y), -0.0625, sin(x)) * fma(sin(x), -0.0625, sin(y))), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

10. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]
11. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot t\_1\right) \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_1 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (*
           (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
           (- (cos x) (cos y)))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.00125)
     (/
      t_0
      (*
       (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) (+ (* (* 0.5 t_1) (cos y)) 1.0))
       3.0))
     (if (<= x 6.6e-24)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos x) t_2 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y))) 3.0))
       (/ t_0 (fma 1.5 (fma (cos x) t_2 (* t_1 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.00125) {
		tmp = t_0 / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (((0.5 * t_1) * cos(y)) + 1.0)) * 3.0);
	} else if (x <= 6.6e-24) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_2, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	} else {
		tmp = t_0 / fma(1.5, fma(cos(x), t_2, (t_1 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.00125)
		tmp = Float64(t_0 / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(Float64(0.5 * t_1) * cos(y)) + 1.0)) * 3.0));
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_2, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(x), t_2, Float64(t_1 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 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] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.00125], N[(t$95$0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-24], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00125000000000000003

   1. Initial program 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 61.0

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

   7. Applied rewrites 61.0%

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

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. lower-sqrt.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   if 6.59999999999999968e-24 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 67.1

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

   8. Applied rewrites 67.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (+
           (*
            (* (- (sin y) (/ (sin x) 16.0)) (* (sin x) (sqrt 2.0)))
            (- (cos x) (cos y)))
           2.0)
          (fma 1.5 (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))))
   (if (<= x -0.00125)
     t_1
     (if (<= x 6.6e-24)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((((sin(y) - (sin(x) / 16.0)) * (sin(x) * sqrt(2.0))) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = t_1;
	} else if (x <= 6.6e-24) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) * sqrt(2.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.00125)
		tmp = t_1;
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00125], t$95$1, If[LessEqual[x, 6.6e-24], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00125000000000000003 or 6.59999999999999968e-24 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sqrt.f64 64.0

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

   8. Applied rewrites 64.0%

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

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. lower-sqrt.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := {\sin y}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;y \leq -0.01:\\ \;\;\;\;\frac{\left(\left(t\_1 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_0 + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_2 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.014:\\ \;\;\;\;\frac{1}{\frac{3}{\mathsf{fma}\left(t\_0, \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), t\_2, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot t\_2, \cos y, 1\right), 3, \frac{3}{\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)} \cdot \cos x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= y -0.01)
     (/
      (+ (* (* (* t_1 -0.0625) (sqrt 2.0)) t_0) 2.0)
      (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_2 (cos y))) 3.0))
     (if (<= y 0.014)
       (/
        1.0
        (*
         (/
          3.0
          (fma
           t_0
           (*
            (* (fma -0.0625 y (sin x)) (sqrt 2.0))
            (fma (sin x) -0.0625 (sin y)))
           2.0))
         (fma
          (fma 0.5 (sqrt 5.0) -0.5)
          (cos x)
          (fma (fma (* -0.25 y) y 0.5) t_2 1.0))))
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_1 2.0)
        (fma
         (fma (* 0.5 t_2) (cos y) 1.0)
         3.0
         (* (/ 3.0 (fma 0.5 (sqrt 5.0) 0.5)) (cos x))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (y <= -0.01) {
		tmp = ((((t_1 * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_2 * cos(y))), 3.0);
	} else if (y <= 0.014) {
		tmp = 1.0 / ((3.0 / fma(t_0, ((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0)) * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma(fma((-0.25 * y), y, 0.5), t_2, 1.0)));
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / fma(fma((0.5 * t_2), cos(y), 1.0), 3.0, ((3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (y <= -0.01)
		tmp = Float64(Float64(Float64(Float64(Float64(t_1 * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_2 * cos(y))), 3.0));
	elseif (y <= 0.014)
		tmp = Float64(1.0 / Float64(Float64(3.0 / fma(t_0, Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * fma(sin(x), -0.0625, sin(y))), 2.0)) * fma(fma(0.5, sqrt(5.0), -0.5), cos(x), fma(fma(Float64(-0.25 * y), y, 0.5), t_2, 1.0))));
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_1, 2.0) / fma(fma(Float64(0.5 * t_2), cos(y), 1.0), 3.0, Float64(Float64(3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x))));
	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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.01], N[(N[(N[(N[(N[(t$95$1 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.014], N[(1.0 / N[(N[(3.0 / N[(t$95$0 * N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(-0.25 * y), $MachinePrecision] * y + 0.5), $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * t$95$2), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(3.0 / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0100000000000000002

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -0.0100000000000000002 < y < 0.0140000000000000003

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 99.4

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

   10. Applied rewrites 99.4%

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

   if 0.0140000000000000003 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-*r/ N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 61.7

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

   9. Applied rewrites 61.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.01:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.014:\\ \;\;\;\;\frac{1}{\frac{3}{\mathsf{fma}\left(\cos x - \cos y, \left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right), 2\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), 3 - \sqrt{5}, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \frac{3}{\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)} \cdot \cos x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := 0.5 \cdot t\_1\\ \mathbf{if}\;y \leq -0.0058:\\ \;\;\;\;\frac{\left(\left(t\_0 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_1 \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0115:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, y\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, t\_2 \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \cos y, 1\right), 3, \frac{3}{\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)} \cdot \cos x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0)) (t_1 (- 3.0 (sqrt 5.0))) (t_2 (* 0.5 t_1)))
   (if (<= y -0.0058)
     (/
      (+ (* (* (* t_0 -0.0625) (sqrt 2.0)) (- (cos x) (cos y))) 2.0)
      (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_1 (cos y))) 3.0))
     (if (<= y 0.0115)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (fma -0.0625 (sin x) y) (* (- (cos x) 1.0) (sqrt 2.0)))
         2.0)
        (*
         (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) (+ (* t_2 (cos y)) 1.0))
         3.0))
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_0 2.0)
        (fma
         (fma t_2 (cos y) 1.0)
         3.0
         (* (/ 3.0 (fma 0.5 (sqrt 5.0) 0.5)) (cos x))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 0.5 * t_1;
	double tmp;
	if (y <= -0.0058) {
		tmp = ((((t_0 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_1 * cos(y))), 3.0);
	} else if (y <= 0.0115) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (fma(-0.0625, sin(x), y) * ((cos(x) - 1.0) * sqrt(2.0))), 2.0) / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), ((t_2 * cos(y)) + 1.0)) * 3.0);
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_0, 2.0) / fma(fma(t_2, cos(y), 1.0), 3.0, ((3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(0.5 * t_1)
	tmp = 0.0
	if (y <= -0.0058)
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_1 * cos(y))), 3.0));
	elseif (y <= 0.0115)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(fma(-0.0625, sin(x), y) * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), 2.0) / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(t_2 * cos(y)) + 1.0)) * 3.0));
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_0, 2.0) / fma(fma(t_2, cos(y), 1.0), 3.0, Float64(Float64(3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x))));
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * t$95$1), $MachinePrecision]}, If[LessEqual[y, -0.0058], N[(N[(N[(N[(N[(t$95$0 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0115], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(t$95$2 * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(3.0 / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0058

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -0.0058 < y < 0.0115

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

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

      1. associate-*l* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if 0.0115 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-*r/ N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 61.7

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

   9. Applied rewrites 61.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0058:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0115:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \mathsf{fma}\left(-0.0625, \sin x, y\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \frac{3}{\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)} \cdot \cos x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot t\_1\right) \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, t\_1 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (* (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)) (- (cos x) (cos y)))
          2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0)))
   (if (<= x -0.00125)
     (/
      t_0
      (*
       (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) (+ (* (* 0.5 t_1) (cos y)) 1.0))
       3.0))
     (if (<= x 6.6e-24)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos x) t_2 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y))) 3.0))
       (/ t_0 (fma 1.5 (fma (cos x) t_2 (* t_1 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = (((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) - 1.0;
	double tmp;
	if (x <= -0.00125) {
		tmp = t_0 / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (((0.5 * t_1) * cos(y)) + 1.0)) * 3.0);
	} else if (x <= 6.6e-24) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_2, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	} else {
		tmp = t_0 / fma(1.5, fma(cos(x), t_2, (t_1 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) - 1.0)
	tmp = 0.0
	if (x <= -0.00125)
		tmp = Float64(t_0 / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(Float64(0.5 * t_1) * cos(y)) + 1.0)) * 3.0));
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_2, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(x), t_2, Float64(t_1 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.00125], N[(t$95$0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-24], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00125000000000000003

   1. Initial program 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 57.3

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

   7. Applied rewrites 57.3%

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

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. lower-sqrt.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   if 6.59999999999999968e-24 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 63.8

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

   8. Applied rewrites 63.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (/
          (+
           (* (* (* (pow (sin x) 2.0) -0.0625) (sqrt 2.0)) (- (cos x) (cos y)))
           2.0)
          (fma 1.5 (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))))
   (if (<= x -0.00125)
     t_1
     (if (<= x 6.6e-24)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = ((((pow(sin(x), 2.0) * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = t_1;
	} else if (x <= 6.6e-24) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(Float64(Float64(Float64(Float64((sin(x) ^ 2.0) * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.00125)
		tmp = t_1;
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00125], t$95$1, If[LessEqual[x, 6.6e-24], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00125000000000000003 or 6.59999999999999968e-24 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 60.5

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

   8. Applied rewrites 60.5%

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

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. lower-sqrt.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\sin x}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_1 \cdot \cos y\right), 3\right)\\ \mathbf{if}\;y \leq -0.00098:\\ \;\;\;\;\frac{\left(\left(t\_0 \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{t\_2}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, t\_0, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot t\_1, \cos y, 1\right), 3, \frac{3}{\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)} \cdot \cos x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_1 (cos y))) 3.0)))
   (if (<= y -0.00098)
     (/ (+ (* (* (* t_0 -0.0625) (sqrt 2.0)) (- (cos x) (cos y))) 2.0) t_2)
     (if (<= y 0.0008)
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
        t_2)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) t_0 2.0)
        (fma
         (fma (* 0.5 t_1) (cos y) 1.0)
         3.0
         (* (/ 3.0 (fma 0.5 (sqrt 5.0) 0.5)) (cos x))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_1 * cos(y))), 3.0);
	double tmp;
	if (y <= -0.00098) {
		tmp = ((((t_0 * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / t_2;
	} else if (y <= 0.0008) {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / t_2;
	} else {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_0, 2.0) / fma(fma((0.5 * t_1), cos(y), 1.0), 3.0, ((3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_1 * cos(y))), 3.0)
	tmp = 0.0
	if (y <= -0.00098)
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / t_2);
	elseif (y <= 0.0008)
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / t_2);
	else
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), t_0, 2.0) / fma(fma(Float64(0.5 * t_1), cos(y), 1.0), 3.0, Float64(Float64(3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x))));
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[y, -0.00098], N[(N[(N[(N[(N[(t$95$0 * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 0.0008], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(3.0 / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.7999999999999997e-4

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if -9.7999999999999997e-4 < y < 8.00000000000000038e-4

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 8.00000000000000038e-4 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-*r/ N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 61.7

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

   9. Applied rewrites 61.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00098:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \frac{3}{\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)} \cdot \cos x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin y}^{2}\\ t_2 := \mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}\\ \mathbf{if}\;y \leq -0.0011:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, t\_2, 2\right)}{3}}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right)}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_1, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot t\_0, \cos y, 1\right), 3, \frac{3}{\mathsf{fma}\left(0.5, \sqrt{5}, 0.5\right)} \cdot \cos x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin y) 2.0))
        (t_2 (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))))
   (if (<= y -0.0011)
     (/
      (/ (fma t_1 t_2 2.0) 3.0)
      (fma (* 0.5 (cos y)) t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0)))
     (if (<= y 0.0008)
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
        (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_0 (cos y))) 3.0))
       (/
        (fma t_2 t_1 2.0)
        (fma
         (fma (* 0.5 t_0) (cos y) 1.0)
         3.0
         (* (/ 3.0 (fma 0.5 (sqrt 5.0) 0.5)) (cos x))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = fma(0.0625, cos(y), -0.0625) * sqrt(2.0);
	double tmp;
	if (y <= -0.0011) {
		tmp = (fma(t_1, t_2, 2.0) / 3.0) / fma((0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0));
	} else if (y <= 0.0008) {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_0 * cos(y))), 3.0);
	} else {
		tmp = fma(t_2, t_1, 2.0) / fma(fma((0.5 * t_0), cos(y), 1.0), 3.0, ((3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0))
	tmp = 0.0
	if (y <= -0.0011)
		tmp = Float64(Float64(fma(t_1, t_2, 2.0) / 3.0) / fma(Float64(0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)));
	elseif (y <= 0.0008)
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(fma(t_2, t_1, 2.0) / fma(fma(Float64(0.5 * t_0), cos(y), 1.0), 3.0, Float64(Float64(3.0 / fma(0.5, sqrt(5.0), 0.5)) * cos(x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0011], N[(N[(N[(t$95$1 * t$95$2 + 2.0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0008], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * t$95$1 + 2.0), $MachinePrecision] / N[(N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Cos[y], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(3.0 / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00110000000000000007

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 61.9%

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

   if -0.00110000000000000007 < y < 8.00000000000000038e-4

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 8.00000000000000038e-4 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 61.6%

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

   7. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-*r/ N/A

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

      5. swap-sqr N/A

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

      6. metadata-eval N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 61.7

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

   9. Applied rewrites 61.7%

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

Alternative 12: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{t\_1}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1
         (fma
          1.5
          (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y)))
          3.0))
        (t_2 (pow (sin x) 2.0)))
   (if (<= x -0.00125)
     (/ (fma (* t_2 -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0) t_1)
     (if (<= x 6.6e-24)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        t_1)
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_2 2.0)
        (fma 1.5 (fma (cos x) t_0 (* (- 3.0 (sqrt 5.0)) (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = fma((t_2 * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_1;
	} else if (x <= 6.6e-24) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_1;
	} else {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_2, 2.0) / fma(1.5, fma(cos(x), t_0, ((3.0 - sqrt(5.0)) * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -0.00125)
		tmp = Float64(fma(Float64(t_2 * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / t_1);
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_1);
	else
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_2, 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(3.0 - sqrt(5.0)) * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -0.00125], N[(N[(N[(t$95$2 * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 6.6e-24], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00125000000000000003

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 99.2%

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

   9. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. lower-sqrt.f64 57.2

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   12. Applied rewrites 57.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. lower-sqrt.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   if 6.59999999999999968e-24 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 63.7

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

   8. Applied rewrites 63.7%

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

Alternative 13: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := {\sin x}^{2}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1.5 \cdot \cos y\right) \cdot t\_2 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_2 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00125)
     (/
      (fma (* t_1 -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
      (fma 1.5 (fma (cos x) t_0 (* (/ 4.0 (+ (sqrt 5.0) 3.0)) (cos y))) 3.0))
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (+
         (* (* 1.5 (cos y)) t_2)
         (fma (fma (sqrt 5.0) 1.5 -1.5) (cos x) 3.0)))
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_1 2.0)
        (fma 1.5 (fma (cos x) t_0 (* t_2 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = pow(sin(x), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = fma((t_1 * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, ((4.0 / (sqrt(5.0) + 3.0)) * cos(y))), 3.0);
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (((1.5 * cos(y)) * t_2) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0));
	} else {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_1, 2.0) / fma(1.5, fma(cos(x), t_0, (t_2 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00125)
		tmp = Float64(fma(Float64(t_1 * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(Float64(4.0 / Float64(sqrt(5.0) + 3.0)) * cos(y))), 3.0));
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(Float64(Float64(1.5 * cos(y)) * t_2) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0)));
	else
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_1, 2.0) / fma(1.5, fma(cos(x), t_0, Float64(t_2 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00125], N[(N[(N[(t$95$1 * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + -1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(t$95$2 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00125000000000000003

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   7. Applied rewrites 99.2%

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

   9. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \sqrt{2}, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\sin y, -0.0625, \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower--.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. lower-sqrt.f64 57.2

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   12. Applied rewrites 57.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)} \]

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   if 6.59999999999999968e-24 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 63.7

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

   8. Applied rewrites 63.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \frac{4}{\sqrt{5} + 3} \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot t\_1\right) \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1.5 \cdot \cos y\right) \cdot t\_1 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_1 \cdot \cos y\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
          (pow (sin x) 2.0)
          2.0))
        (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00125)
     (/
      t_0
      (*
       (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) (+ (* (* 0.5 t_1) (cos y)) 1.0))
       3.0))
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (+
         (* (* 1.5 (cos y)) t_1)
         (fma (fma (sqrt 5.0) 1.5 -1.5) (cos x) 3.0)))
       (/
        t_0
        (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_1 (cos y))) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = t_0 / (fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), (((0.5 * t_1) * cos(y)) + 1.0)) * 3.0);
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (((1.5 * cos(y)) * t_1) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0));
	} else {
		tmp = t_0 / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_1 * cos(y))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00125)
		tmp = Float64(t_0 / Float64(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), Float64(Float64(Float64(0.5 * t_1) * cos(y)) + 1.0)) * 3.0));
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(Float64(Float64(1.5 * cos(y)) * t_1) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0)));
	else
		tmp = Float64(t_0 / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_1 * cos(y))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00125], N[(t$95$0 / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[(N[(0.5 * t$95$1), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + -1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00125000000000000003

   1. Initial program 99.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-sub N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 99.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \color{blue}{\cos x}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \color{blue}{\sqrt{2}}, {\sin x}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, \color{blue}{{\sin x}^{2}}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \cos y \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

      16. lower-sin.f64 57.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\color{blue}{\sin x}}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   7. Applied rewrites 57.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)} \]

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right) + \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)}} \]

   if 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \color{blue}{\cos x}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \color{blue}{\sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, \color{blue}{{\sin x}^{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      16. lower-sin.f64 63.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\color{blue}{\sin x}}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 63.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot \cos y + 1\right) \cdot 3}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)\\ \mathbf{if}\;y \leq -0.00098:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(0.5 \cdot \cos y, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\left(1.5 \cdot \cos y\right) \cdot t\_0 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0))
          (pow (sin y) 2.0)
          2.0)))
   (if (<= y -0.00098)
     (/
      t_1
      (*
       (fma (* 0.5 (cos y)) t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
       3.0))
     (if (<= y 0.0008)
       (/
        (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
        (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_0 (cos y))) 3.0))
       (/
        t_1
        (+
         (* (* 1.5 (cos y)) t_0)
         (fma (fma (sqrt 5.0) 1.5 -1.5) (cos x) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0);
	double tmp;
	if (y <= -0.00098) {
		tmp = t_1 / (fma((0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
	} else if (y <= 0.0008) {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_0 * cos(y))), 3.0);
	} else {
		tmp = t_1 / (((1.5 * cos(y)) * t_0) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0)
	tmp = 0.0
	if (y <= -0.00098)
		tmp = Float64(t_1 / Float64(fma(Float64(0.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0));
	elseif (y <= 0.0008)
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_0 * cos(y))), 3.0));
	else
		tmp = Float64(t_1 / Float64(Float64(Float64(1.5 * cos(y)) * t_0) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[y, -0.00098], N[(t$95$1 / N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0008], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + -1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.7999999999999997e-4

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 61.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]

   if -9.7999999999999997e-4 < y < 8.00000000000000038e-4

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \color{blue}{\cos x}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \color{blue}{\sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, \color{blue}{{\sin x}^{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      16. lower-sin.f64 98.5

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\color{blue}{\sin x}}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if 8.00000000000000038e-4 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 61.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

   9. Applied rewrites 61.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right) + \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00098:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{elif}\;y \leq 0.0008:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1.5 \cdot \cos y\right) \cdot t\_0 + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_0 (cos y))) 3.0))))
   (if (<= x -0.00125)
     t_1
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (+
         (* (* 1.5 (cos y)) t_0)
         (fma (fma (sqrt 5.0) 1.5 -1.5) (cos x) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_0 * cos(y))), 3.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = t_1;
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / (((1.5 * cos(y)) * t_0) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_0 * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.00125)
		tmp = t_1;
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / Float64(Float64(Float64(1.5 * cos(y)) * t_0) + fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00125], t$95$1, If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + -1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00125000000000000003 or 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \color{blue}{\cos x}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \color{blue}{\sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, \color{blue}{{\sin x}^{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      16. lower-sin.f64 60.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\color{blue}{\sin x}}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right) + \left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(1.5 \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right) + \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -0.00125:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_0 (cos y))) 3.0))))
   (if (<= x -0.00125)
     t_1
     (if (<= x 6.6e-24)
       (/
        (fma (pow (sin y) 2.0) (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) 2.0)
        (fma (* 1.5 (cos y)) t_0 (fma (fma (sqrt 5.0) 1.5 -1.5) (cos x) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_0 * cos(y))), 3.0);
	double tmp;
	if (x <= -0.00125) {
		tmp = t_1;
	} else if (x <= 6.6e-24) {
		tmp = fma(pow(sin(y), 2.0), (fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), 2.0) / fma((1.5 * cos(y)), t_0, fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_0 * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -0.00125)
		tmp = t_1;
	elseif (x <= 6.6e-24)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(fma(sqrt(5.0), 1.5, -1.5), cos(x), 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00125], t$95$1, If[LessEqual[x, 6.6e-24], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + -1.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00125000000000000003 or 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \color{blue}{\cos x}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \color{blue}{\sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, \color{blue}{{\sin x}^{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      16. lower-sin.f64 60.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\color{blue}{\sin x}}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -0.00125000000000000003 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin y}^{2}, \mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 1.5, -1.5\right), \cos x, 3\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0 \cdot \cos y\right), 3\right)}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), 3, 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (* t_0 (cos y))) 3.0))))
   (if (<= x -3.4e-6)
     t_1
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma (* 1.5 (cos y)) t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) 3.0 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (t_0 * cos(y))), 3.0);
	double tmp;
	if (x <= -3.4e-6) {
		tmp = t_1;
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma((1.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), 3.0, 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(t_0 * cos(y))), 3.0))
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = t_1;
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), 3.0, 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(t$95$0 * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], t$95$1, If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000006e-6 or 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}\right)} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot {\sin x}^{2}} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2} + 2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right)\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      10. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right)} \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \color{blue}{\cos x}, \frac{1}{16}\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \color{blue}{\sqrt{2}}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      15. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \cos x, \frac{1}{16}\right) \cdot \sqrt{2}, \color{blue}{{\sin x}^{2}}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

      16. lower-sin.f64 60.4

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\color{blue}{\sin x}}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \left(3 - \sqrt{5}\right) \cdot \cos y\right), 3\right)} \]

   if -3.40000000000000006e-6 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3 + \left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right)} \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \cos x\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \frac{-1}{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5}} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3\right)\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 + \left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3}} \]

       2. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot 3} + 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}, 3, 3\right)}\right)} \]

       11. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 3, 3\right)\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}, 3, 3\right)\right)} \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}, 3, 3\right)\right)} \]

       14. lower-sqrt.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right), 3, 3\right)\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), 3, 3\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0\right), 1\right)}{t\_1} \cdot 3}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(t\_2, 3, 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(3 \cdot \cos x, t\_2, \mathsf{fma}\left(1.5, t\_0, 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          (* (pow (sin x) 2.0) -0.0625)
          (* (- (cos x) 1.0) (sqrt 2.0))
          2.0))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= x -3.4e-6)
     (/ 1.0 (* (/ (fma 0.5 (fma (cos x) (- (sqrt 5.0) 1.0) t_0) 1.0) t_1) 3.0))
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma (* 1.5 (cos y)) t_0 (fma t_2 3.0 3.0)))
       (/ t_1 (fma (* 3.0 (cos x)) t_2 (fma 1.5 t_0 3.0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((pow(sin(x), 2.0) * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (x <= -3.4e-6) {
		tmp = 1.0 / ((fma(0.5, fma(cos(x), (sqrt(5.0) - 1.0), t_0), 1.0) / t_1) * 3.0);
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma((1.5 * cos(y)), t_0, fma(t_2, 3.0, 3.0));
	} else {
		tmp = t_1 / fma((3.0 * cos(x)), t_2, fma(1.5, t_0, 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64((sin(x) ^ 2.0) * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0)
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = Float64(1.0 / Float64(Float64(fma(0.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), t_0), 1.0) / t_1) * 3.0));
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(t_2, 3.0, 3.0)));
	else
		tmp = Float64(t_1 / fma(Float64(3.0 * cos(x)), t_2, fma(1.5, t_0, 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], N[(1.0 / N[(N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(t$95$2 * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(3.0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(1.5 * t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000006e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{-1}{4} \cdot \left({y}^{2} \cdot \left(3 - \sqrt{5}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

   5. Applied rewrites 46.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) + 1\right)}} \]

   7. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), 3 - \sqrt{5}, 1\right)\right) \cdot \frac{3}{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \sqrt{2}\right), 2\right)}}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} \cdot 3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)} \cdot 3}} \]

   10. Applied rewrites 55.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 1\right)}{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)} \cdot 3}} \]

   if -3.40000000000000006e-6 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3 + \left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right)} \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \cos x\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \frac{-1}{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5}} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3\right)\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 + \left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3}} \]

       2. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot 3} + 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}, 3, 3\right)}\right)} \]

       11. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 3, 3\right)\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}, 3, 3\right)\right)} \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}, 3, 3\right)\right)} \]

       14. lower-sqrt.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right), 3, 3\right)\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), 3, 3\right)\right)}} \]

   if 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 29.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 29.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3 + \left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right)} \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \cos x\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \frac{-1}{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5}} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

   9. Applied rewrites 29.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3\right)\right)}} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       7. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       8. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       11. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       12. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       13. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

   12. Applied rewrites 62.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(3 \cdot \cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \mathsf{fma}\left(1.5, 3 - \sqrt{5}, 3\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, t\_0, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_1\right), 3\right)}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_1, \mathsf{fma}\left(t\_2, 3, 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(3 \cdot \cos x, t\_2, \mathsf{fma}\left(1.5, t\_1, 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma 0.5 (sqrt 5.0) -0.5)))
   (if (<= x -3.4e-6)
     (/
      (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) t_0 2.0)
      (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) t_1) 3.0))
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma (* 1.5 (cos y)) t_1 (fma t_2 3.0 3.0)))
       (/
        (fma (* t_0 -0.0625) (* (- (cos x) 1.0) (sqrt 2.0)) 2.0)
        (fma (* 3.0 (cos x)) t_2 (fma 1.5 t_1 3.0)))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(0.5, sqrt(5.0), -0.5);
	double tmp;
	if (x <= -3.4e-6) {
		tmp = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), t_1), 3.0);
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma((1.5 * cos(y)), t_1, fma(t_2, 3.0, 3.0));
	} else {
		tmp = fma((t_0 * -0.0625), ((cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma((3.0 * cos(x)), t_2, fma(1.5, t_1, 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = fma(0.5, sqrt(5.0), -0.5)
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), t_1), 3.0));
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(Float64(1.5 * cos(y)), t_1, fma(t_2, 3.0, 3.0)));
	else
		tmp = Float64(fma(Float64(t_0 * -0.0625), Float64(Float64(cos(x) - 1.0) * sqrt(2.0)), 2.0) / fma(Float64(3.0 * cos(x)), t_2, fma(1.5, t_1, 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(t$95$2 * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * -0.0625), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(3.0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(1.5 * t$95$1 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000006e-6

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 26.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 26.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 55.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

   if -3.40000000000000006e-6 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3 + \left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right)} \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \cos x\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \frac{-1}{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5}} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3\right)\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 + \left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3}} \]

       2. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot 3} + 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}, 3, 3\right)}\right)} \]

       11. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 3, 3\right)\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}, 3, 3\right)\right)} \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}, 3, 3\right)\right)} \]

       14. lower-sqrt.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right), 3, 3\right)\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), 3, 3\right)\right)}} \]

   if 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 29.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 29.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3 + \left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right)} \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \cos x\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \frac{-1}{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5}} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

   9. Applied rewrites 29.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3\right)\right)}} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)} + 2}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin x}^{2}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2} \cdot \frac{-1}{16}}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       7. lower-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin x}^{2}} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       8. lower-sin.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin x}}^{2} \cdot \frac{-1}{16}, \sqrt{2} \cdot \left(\cos x - 1\right), 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right) \cdot \sqrt{2}}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       11. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \color{blue}{\left(\cos x - 1\right)} \cdot \sqrt{2}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       12. lower-cos.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\color{blue}{\cos x} - 1\right) \cdot \sqrt{2}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

       13. lower-sqrt.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \frac{-1}{16}, \left(\cos x - 1\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \left(\frac{3}{2} \cdot \left(3 - \sqrt{5}\right) + 3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)\right)} \]

   12. Applied rewrites 62.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\sin x}^{2} \cdot -0.0625, \left(\cos x - 1\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(3 \cdot \cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \mathsf{fma}\left(1.5, 3 - \sqrt{5}, 3\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), 3, 3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) t_0) 3.0))))
   (if (<= x -3.4e-6)
     t_1
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma (* 1.5 (cos y)) t_0 (fma (fma 0.5 (sqrt 5.0) -0.5) 3.0 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), t_0), 3.0);
	double tmp;
	if (x <= -3.4e-6) {
		tmp = t_1;
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma((1.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), 3.0, 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), t_0), 3.0))
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = t_1;
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(Float64(1.5 * cos(y)), t_0, fma(fma(0.5, sqrt(5.0), -0.5), 3.0, 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], t$95$1, If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(1.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000006e-6 or 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 27.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 27.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 59.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

   if -3.40000000000000006e-6 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3 + \left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) \cdot \cos x} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right)} \cdot \cos x + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \cos x\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\left(\color{blue}{\sqrt{5} \cdot \frac{1}{2}} + \frac{-1}{2}\right) \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      8. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\color{blue}{\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)} \cdot \cos x\right) + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2} + \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5}} + \frac{-1}{2}\right) \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)} \cdot \cos x\right) \cdot 3 + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right) \cdot \left(\cos x \cdot 3\right)} + \mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right) \cdot 3} \]

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x \cdot 3, \mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, 3\right)\right)}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 + \left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3}} \]

       2. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(\frac{3}{2} \cdot \cos y\right) \cdot \left(3 - \sqrt{5}\right)} + \left(3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)}} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2} \cdot \cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       6. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \color{blue}{\cos y}, 3 - \sqrt{5}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, \color{blue}{3 - \sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       8. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \color{blue}{\sqrt{5}}, 3 \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + 3\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot 3} + 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}, 3, 3\right)}\right)} \]

       11. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 3, 3\right)\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}, 3, 3\right)\right)} \]

       13. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2} \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}, 3, 3\right)\right)} \]

       14. lower-sqrt.f64 99.6

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right), 3, 3\right)\right)} \]

   12. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), 3, 3\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma (cos x) t_0 t_1) 3.0))))
   (if (<= x -3.4e-6)
     t_2
     (if (<= x 6.6e-24)
       (/
        (fma (* (fma 0.0625 (cos y) -0.0625) (sqrt 2.0)) (pow (sin y) 2.0) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0);
	double tmp;
	if (x <= -3.4e-6) {
		tmp = t_2;
	} else if (x <= 6.6e-24) {
		tmp = fma((fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), pow(sin(y), 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0))
	tmp = 0.0
	if (x <= -3.4e-6)
		tmp = t_2;
	elseif (x <= 6.6e-24)
		tmp = Float64(fma(Float64(fma(0.0625, cos(y), -0.0625) * sqrt(2.0)), (sin(y) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-6], t$95$2, If[LessEqual[x, 6.6e-24], N[(N[(N[(N[(0.0625 * N[Cos[y], $MachinePrecision] + -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000006e-6 or 6.59999999999999968e-24 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 27.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. Applied rewrites 27.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   10. Applied rewrites 59.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

   if -3.40000000000000006e-6 < x < 6.59999999999999968e-24

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} - 1}\right), 3\right)} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} - 1\right), 3\right)} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma (* (fma -0.0625 (cos x) 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
  (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return fma((fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(fma(-0.0625, cos(x), 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

7. Applied rewrites 62.6%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), \cos y, 1\right), 3, \left(3 \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) \cdot \cos x\right)}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 \cdot \left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3 \cdot \left(1 + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

10. Applied rewrites 58.8%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \cos x, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]
11. Add Preprocessing

Alternative 24: 42.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (cos x) (- (sqrt 5.0) 1.0) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(x), (sqrt(5.0) - 1.0), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(x), Float64(sqrt(5.0) - 1.0), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{16} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot {\sin y}^{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot {\sin y}^{2}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-1}{16} \cdot \color{blue}{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)}\right) \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} \cdot {\sin y}^{2} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 62.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   4. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(3 \cdot \frac{1}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} - 1}, 3 - \sqrt{5}\right), 3\right)} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{16}, \cos y, \frac{-1}{16}\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   13. lower-sqrt.f64 41.9

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 41.9%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0625, \cos y, -0.0625\right) \cdot \sqrt{2}, {\sin y}^{2}, 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{2 + \color{blue}{\frac{-1}{32} \cdot \left({y}^{4} \cdot \sqrt{2}\right)}}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 31.2%

    \[\leadsto \frac{\mathsf{fma}\left(-0.03125 \cdot {y}^{4}, \color{blue}{\sqrt{2}}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
13. Step-by-step derivation

14. Applied rewrites 42.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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))))))