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

Percentage Accurate: 99.3% → 99.4%

Time: 25.3s

Alternatives: 37

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

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (fma (sin y) -0.0625 (sin x))
   (* (fma -0.0625 (sin x) (sin y)) (* (- (cos x) (cos y)) (sqrt 2.0)))
   2.0)
  (fma
   (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
   3.0
   (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0)))))

C

double code(double x, double y) {
	return fma(fma(sin(y), -0.0625, sin(x)), (fma(-0.0625, sin(x), sin(y)) * ((cos(x) - cos(y)) * sqrt(2.0))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
}

Julia

function code(x, y)
	return Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(Float64(cos(x) - cos(y)) * sqrt(2.0))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $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] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 99.5

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (fma (sin y) -0.0625 (sin x))
   (* (sqrt 2.0) (* (- (cos x) (cos y)) (fma (sin x) -0.0625 (sin y))))
   2.0)
  (fma
   (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
   3.0
   (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0)))))

C

double code(double x, double y) {
	return fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * ((cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y)))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
}

Julia

function code(x, y)
	return Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * fma(sin(x), -0.0625, sin(y)))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. div-sub N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in y around inf

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 6: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + 3\\ t_1 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_2 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_3 := \mathsf{fma}\left(t\_2, \cos x, 1\right)\\ t_4 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(t\_4 \cdot t\_1\right), 2\right)}{\mathsf{fma}\left(t\_3, 3, \frac{\cos y \cdot 6}{t\_0}\right)}\\ \mathbf{elif}\;x \leq 0.25:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right)}{\mathsf{fma}\left(t\_3, 3, \left(\cos y \cdot 3\right) \cdot \frac{2}{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_4, \sqrt{2} \cdot \left(t\_1 \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_2, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) 3.0))
        (t_1 (fma (sin x) -0.0625 (sin y)))
        (t_2 (fma (sqrt 5.0) 0.5 -0.5))
        (t_3 (fma t_2 (cos x) 1.0))
        (t_4 (- (cos x) (cos y))))
   (if (<= x -0.15)
     (/
      (fma (sin x) (* (sqrt 2.0) (* t_4 t_1)) 2.0)
      (fma t_3 3.0 (/ (* (cos y) 6.0) t_0)))
     (if (<= x 0.25)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ (sin x) 16.0)))
          (fma
           x
           (* x (fma (* x x) 0.041666666666666664 -0.5))
           (- 1.0 (cos y)))))
        (fma t_3 3.0 (* (* (cos y) 3.0) (/ 2.0 t_0))))
       (*
        (fma t_4 (* (sqrt 2.0) (* t_1 (sin x))) 2.0)
        (/
         0.3333333333333333
         (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_2 1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + 3.0;
	double t_1 = fma(sin(x), -0.0625, sin(y));
	double t_2 = fma(sqrt(5.0), 0.5, -0.5);
	double t_3 = fma(t_2, cos(x), 1.0);
	double t_4 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.15) {
		tmp = fma(sin(x), (sqrt(2.0) * (t_4 * t_1)), 2.0) / fma(t_3, 3.0, ((cos(y) * 6.0) / t_0));
	} else if (x <= 0.25) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * fma(x, (x * fma((x * x), 0.041666666666666664, -0.5)), (1.0 - cos(y))))) / fma(t_3, 3.0, ((cos(y) * 3.0) * (2.0 / t_0)));
	} else {
		tmp = fma(t_4, (sqrt(2.0) * (t_1 * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_2, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + 3.0)
	t_1 = fma(sin(x), -0.0625, sin(y))
	t_2 = fma(sqrt(5.0), 0.5, -0.5)
	t_3 = fma(t_2, cos(x), 1.0)
	t_4 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.15)
		tmp = Float64(fma(sin(x), Float64(sqrt(2.0) * Float64(t_4 * t_1)), 2.0) / fma(t_3, 3.0, Float64(Float64(cos(y) * 6.0) / t_0)));
	elseif (x <= 0.25)
		tmp = 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))) * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, -0.5)), Float64(1.0 - cos(y))))) / fma(t_3, 3.0, Float64(Float64(cos(y) * 3.0) * Float64(2.0 / t_0))));
	else
		tmp = Float64(fma(t_4, Float64(sqrt(2.0) * Float64(t_1 * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_2, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.15], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.25], 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[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] * N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.149999999999999994

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-sin.f64 56.7

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

   11. Applied rewrites 56.7%

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

   if -0.149999999999999994 < x < 0.25

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. div-sub N/A

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

      7. lift--.f64 N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. rem-square-sqrt N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-cos.f64 99.8

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{2}{3 + \sqrt{5}}\right)} \]

   9. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{2}{3 + \sqrt{5}}\right)} \]

   if 0.25 < x

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.25:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1 - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \frac{2}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)\\ t_3 := \cos x - \cos y\\ t_4 := \sqrt{2} \cdot \left(t\_3 \cdot t\_0\right)\\ \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, t\_4, 2\right)}{t\_2}\\ \mathbf{elif}\;x \leq 0.195:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\right), t\_4, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, \sqrt{2} \cdot \left(t\_0 \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin x) -0.0625 (sin y)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2
         (fma
          (fma t_1 (cos x) 1.0)
          3.0
          (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))
        (t_3 (- (cos x) (cos y)))
        (t_4 (* (sqrt 2.0) (* t_3 t_0))))
   (if (<= x -0.095)
     (/ (fma (sin x) t_4 2.0) t_2)
     (if (<= x 0.195)
       (/
        (fma
         (fma -0.0625 (sin y) (fma -0.16666666666666666 (* x (* x x)) x))
         t_4
         2.0)
        t_2)
       (*
        (fma t_3 (* (sqrt 2.0) (* t_0 (sin x))) 2.0)
        (/
         0.3333333333333333
         (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_1 1.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(x), -0.0625, sin(y));
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = fma(fma(t_1, cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	double t_3 = cos(x) - cos(y);
	double t_4 = sqrt(2.0) * (t_3 * t_0);
	double tmp;
	if (x <= -0.095) {
		tmp = fma(sin(x), t_4, 2.0) / t_2;
	} else if (x <= 0.195) {
		tmp = fma(fma(-0.0625, sin(y), fma(-0.16666666666666666, (x * (x * x)), x)), t_4, 2.0) / t_2;
	} else {
		tmp = fma(t_3, (sqrt(2.0) * (t_0 * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(x), -0.0625, sin(y))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = fma(fma(t_1, cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0)))
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(sqrt(2.0) * Float64(t_3 * t_0))
	tmp = 0.0
	if (x <= -0.095)
		tmp = Float64(fma(sin(x), t_4, 2.0) / t_2);
	elseif (x <= 0.195)
		tmp = Float64(fma(fma(-0.0625, sin(y), fma(-0.16666666666666666, Float64(x * Float64(x * x)), x)), t_4, 2.0) / t_2);
	else
		tmp = Float64(fma(t_3, Float64(sqrt(2.0) * Float64(t_0 * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.095], N[(N[(N[Sin[x], $MachinePrecision] * t$95$4 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 0.195], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * t$95$4 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.095000000000000001

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-sin.f64 56.7

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

   11. Applied rewrites 56.7%

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

   if -0.095000000000000001 < x < 0.19500000000000001

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*l* N/A

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

       6. *-lft-identity N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 99.8

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(x \cdot x\right)} \cdot x, x\right)\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 99.8%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.0625, \sin y, \mathsf{fma}\left(-0.16666666666666666, \left(x \cdot x\right) \cdot x, x\right)\right)}, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

   if 0.19500000000000001 < x

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.095:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.195:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), x\right)\right), \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \mathsf{fma}\left(t\_1, \cos x, 1\right)\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.059:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(t\_3 \cdot t\_0\right), 2\right)}{\mathsf{fma}\left(t\_2, 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.053:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(t\_0 \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(t\_2, 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, \sqrt{2} \cdot \left(t\_0 \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin x) -0.0625 (sin y)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (fma t_1 (cos x) 1.0))
        (t_3 (- (cos x) (cos y))))
   (if (<= x -0.059)
     (/
      (fma (sin x) (* (sqrt 2.0) (* t_3 t_0)) 2.0)
      (fma t_2 3.0 (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))
     (if (<= x 0.053)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (sqrt 2.0) (* t_0 (fma x (* x -0.5) (- 1.0 (cos y)))))
         2.0)
        (fma t_2 3.0 (* (* (cos y) 3.0) (- 1.5 (* (sqrt 5.0) 0.5)))))
       (*
        (fma t_3 (* (sqrt 2.0) (* t_0 (sin x))) 2.0)
        (/
         0.3333333333333333
         (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_1 1.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(x), -0.0625, sin(y));
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = fma(t_1, cos(x), 1.0);
	double t_3 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.059) {
		tmp = fma(sin(x), (sqrt(2.0) * (t_3 * t_0)), 2.0) / fma(t_2, 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	} else if (x <= 0.053) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (t_0 * fma(x, (x * -0.5), (1.0 - cos(y))))), 2.0) / fma(t_2, 3.0, ((cos(y) * 3.0) * (1.5 - (sqrt(5.0) * 0.5))));
	} else {
		tmp = fma(t_3, (sqrt(2.0) * (t_0 * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(x), -0.0625, sin(y))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = fma(t_1, cos(x), 1.0)
	t_3 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.059)
		tmp = Float64(fma(sin(x), Float64(sqrt(2.0) * Float64(t_3 * t_0)), 2.0) / fma(t_2, 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))));
	elseif (x <= 0.053)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(t_0 * fma(x, Float64(x * -0.5), Float64(1.0 - cos(y))))), 2.0) / fma(t_2, 3.0, Float64(Float64(cos(y) * 3.0) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))));
	else
		tmp = Float64(fma(t_3, Float64(sqrt(2.0) * Float64(t_0 * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.059], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$2 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.053], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(x * N[(x * -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$2 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.058999999999999997

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-sin.f64 56.7

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

   11. Applied rewrites 56.7%

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

   if -0.058999999999999997 < x < 0.0529999999999999985

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 0.0529999999999999985 < x

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.059:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.053:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(t\_2 \cdot t\_0\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right)}{3 \cdot \mathsf{fma}\left(t\_1, \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \sqrt{2} \cdot \left(t\_0 \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin x) -0.0625 (sin y)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2 (- (cos x) (cos y))))
   (if (<= x -0.038)
     (/
      (fma (sin x) (* (sqrt 2.0) (* t_2 t_0)) 2.0)
      (fma (fma t_1 (cos x) 1.0) 3.0 (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))
     (if (<= x 0.046)
       (/
        (+
         2.0
         (*
          t_2
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (fma -0.0625 x (sin y)))))
        (*
         3.0
         (fma t_1 (cos x) (+ 1.0 (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))
       (*
        (fma t_2 (* (sqrt 2.0) (* t_0 (sin x))) 2.0)
        (/
         0.3333333333333333
         (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_1 1.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(x), -0.0625, sin(y));
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.038) {
		tmp = fma(sin(x), (sqrt(2.0) * (t_2 * t_0)), 2.0) / fma(fma(t_1, cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	} else if (x <= 0.046) {
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * fma(-0.0625, x, sin(y))))) / (3.0 * fma(t_1, cos(x), (1.0 + ((3.0 - sqrt(5.0)) * (cos(y) * 0.5)))));
	} else {
		tmp = fma(t_2, (sqrt(2.0) * (t_0 * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(x), -0.0625, sin(y))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.038)
		tmp = Float64(fma(sin(x), Float64(sqrt(2.0) * Float64(t_2 * t_0)), 2.0) / fma(fma(t_1, cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))));
	elseif (x <= 0.046)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * fma(-0.0625, x, sin(y))))) / Float64(3.0 * fma(t_1, cos(x), Float64(1.0 + Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5))))));
	else
		tmp = Float64(fma(t_2, Float64(sqrt(2.0) * Float64(t_0 * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.038], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.046], N[(N[(2.0 + N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + N[(1.0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0379999999999999991

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-sin.f64 56.7

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

   11. Applied rewrites 56.7%

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

   if -0.0379999999999999991 < x < 0.045999999999999999

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 99.4

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \color{blue}{\sin y}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0625, x, \sin y\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   if 0.045999999999999999 < x

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.038:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.046:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)\\ t_3 := \cos x - \cos y\\ t_4 := \sqrt{2} \cdot \left(t\_3 \cdot t\_0\right)\\ \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, t\_4, 2\right)}{t\_2}\\ \mathbf{elif}\;x \leq 0.037:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right), t\_4, 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, \sqrt{2} \cdot \left(t\_0 \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin x) -0.0625 (sin y)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2
         (fma
          (fma t_1 (cos x) 1.0)
          3.0
          (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))
        (t_3 (- (cos x) (cos y)))
        (t_4 (* (sqrt 2.0) (* t_3 t_0))))
   (if (<= x -0.034)
     (/ (fma (sin x) t_4 2.0) t_2)
     (if (<= x 0.037)
       (/ (fma (fma -0.0625 (sin y) x) t_4 2.0) t_2)
       (*
        (fma t_3 (* (sqrt 2.0) (* t_0 (sin x))) 2.0)
        (/
         0.3333333333333333
         (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_1 1.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(x), -0.0625, sin(y));
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = fma(fma(t_1, cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	double t_3 = cos(x) - cos(y);
	double t_4 = sqrt(2.0) * (t_3 * t_0);
	double tmp;
	if (x <= -0.034) {
		tmp = fma(sin(x), t_4, 2.0) / t_2;
	} else if (x <= 0.037) {
		tmp = fma(fma(-0.0625, sin(y), x), t_4, 2.0) / t_2;
	} else {
		tmp = fma(t_3, (sqrt(2.0) * (t_0 * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(x), -0.0625, sin(y))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = fma(fma(t_1, cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0)))
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(sqrt(2.0) * Float64(t_3 * t_0))
	tmp = 0.0
	if (x <= -0.034)
		tmp = Float64(fma(sin(x), t_4, 2.0) / t_2);
	elseif (x <= 0.037)
		tmp = Float64(fma(fma(-0.0625, sin(y), x), t_4, 2.0) / t_2);
	else
		tmp = Float64(fma(t_3, Float64(sqrt(2.0) * Float64(t_0 * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.034], N[(N[(N[Sin[x], $MachinePrecision] * t$95$4 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 0.037], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * t$95$4 + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.034000000000000002

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-sin.f64 56.7

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

   11. Applied rewrites 56.7%

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

   if -0.034000000000000002 < x < 0.0369999999999999982

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sin.f64 99.2

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

   11. Applied rewrites 99.2%

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

   if 0.0369999999999999982 < x

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.034:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.037:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, x\right), \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{2 + t\_1 \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right)}{3 \cdot \mathsf{fma}\left(t\_0, \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{if}\;y \leq -0.003:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_0, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (/
          (+
           2.0
           (* t_1 (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (sin y))))
          (*
           3.0
           (fma t_0 (cos x) (+ 1.0 (* (- 3.0 (sqrt 5.0)) (* (cos y) 0.5))))))))
   (if (<= y -0.003)
     t_2
     (if (<= y 1.2e-6)
       (*
        (fma t_1 (* (sqrt 2.0) (* (fma (sin x) -0.0625 (sin y)) (sin x))) 2.0)
        (/
         0.3333333333333333
         (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_0 1.0))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = cos(x) - cos(y);
	double t_2 = (2.0 + (t_1 * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * sin(y)))) / (3.0 * fma(t_0, cos(x), (1.0 + ((3.0 - sqrt(5.0)) * (cos(y) * 0.5)))));
	double tmp;
	if (y <= -0.003) {
		tmp = t_2;
	} else if (y <= 1.2e-6) {
		tmp = fma(t_1, (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_0, 1.0)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(Float64(2.0 + Float64(t_1 * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * sin(y)))) / Float64(3.0 * fma(t_0, cos(x), Float64(1.0 + Float64(Float64(3.0 - sqrt(5.0)) * Float64(cos(y) * 0.5))))))
	tmp = 0.0
	if (y <= -0.003)
		tmp = t_2;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_0, 1.0))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(t$95$1 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + N[(1.0 + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.003], t$95$2, If[LessEqual[y, 1.2e-6], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0030000000000000001 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

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

      1. lift-+.f64 N/A

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

      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%

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

   5. Taylor expanded in x around 0

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

      1. lower-sin.f64 62.1

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

   7. Applied rewrites 62.1%

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

   if -0.0030000000000000001 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.003:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \sin y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\cos y \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\\ t_1 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(t\_1, \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)\\ t_3 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.0105:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(t\_3 \cdot t\_0\right), 2\right)}{t\_2}\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right), 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, \sqrt{2} \cdot \left(t\_0 \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_1, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin x) -0.0625 (sin y)))
        (t_1 (fma (sqrt 5.0) 0.5 -0.5))
        (t_2
         (fma
          (fma t_1 (cos x) 1.0)
          3.0
          (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))
        (t_3 (- (cos x) (cos y))))
   (if (<= x -0.0105)
     (/ (fma (sin x) (* (sqrt 2.0) (* t_3 t_0)) 2.0) t_2)
     (if (<= x 0.0048)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (sqrt 2.0) (* (- 1.0 (cos y)) (fma -0.0625 x (sin y))))
         2.0)
        t_2)
       (*
        (fma t_3 (* (sqrt 2.0) (* t_0 (sin x))) 2.0)
        (/
         0.3333333333333333
         (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_1 1.0))))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(x), -0.0625, sin(y));
	double t_1 = fma(sqrt(5.0), 0.5, -0.5);
	double t_2 = fma(fma(t_1, cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	double t_3 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.0105) {
		tmp = fma(sin(x), (sqrt(2.0) * (t_3 * t_0)), 2.0) / t_2;
	} else if (x <= 0.0048) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * ((1.0 - cos(y)) * fma(-0.0625, x, sin(y)))), 2.0) / t_2;
	} else {
		tmp = fma(t_3, (sqrt(2.0) * (t_0 * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(x), -0.0625, sin(y))
	t_1 = fma(sqrt(5.0), 0.5, -0.5)
	t_2 = fma(fma(t_1, cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0)))
	t_3 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.0105)
		tmp = Float64(fma(sin(x), Float64(sqrt(2.0) * Float64(t_3 * t_0)), 2.0) / t_2);
	elseif (x <= 0.0048)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * fma(-0.0625, x, sin(y)))), 2.0) / t_2);
	else
		tmp = Float64(fma(t_3, Float64(sqrt(2.0) * Float64(t_0 * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0105], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[x, 0.0048], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0105000000000000007

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. lower-sin.f64 56.7

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

   11. Applied rewrites 56.7%

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

   if -0.0105000000000000007 < x < 0.00479999999999999958

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-out N/A

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

       4. lower-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-sin.f64 99.0

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

   11. Applied rewrites 99.0%

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

   if 0.00479999999999999958 < x

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0105:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin x, \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\\ t_1 := \mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, t\_0, 1\right)\right)}\\ \mathbf{if}\;x \leq -0.0105:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt 5.0) 0.5 -0.5))
        (t_1
         (*
          (fma
           (- (cos x) (cos y))
           (* (sqrt 2.0) (* (fma (sin x) -0.0625 (sin y)) (sin x)))
           2.0)
          (/
           0.3333333333333333
           (fma (cos y) (fma (sqrt 5.0) -0.5 1.5) (fma (cos x) t_0 1.0))))))
   (if (<= x -0.0105)
     t_1
     (if (<= x 0.0048)
       (/
        (fma
         (fma (sin y) -0.0625 (sin x))
         (* (sqrt 2.0) (* (- 1.0 (cos y)) (fma -0.0625 x (sin y))))
         2.0)
        (fma (fma t_0 (cos x) 1.0) 3.0 (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(5.0), 0.5, -0.5);
	double t_1 = fma((cos(x) - cos(y)), (sqrt(2.0) * (fma(sin(x), -0.0625, sin(y)) * sin(x))), 2.0) * (0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_0, 1.0)));
	double tmp;
	if (x <= -0.0105) {
		tmp = t_1;
	} else if (x <= 0.0048) {
		tmp = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * ((1.0 - cos(y)) * fma(-0.0625, x, sin(y)))), 2.0) / fma(fma(t_0, cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(5.0), 0.5, -0.5)
	t_1 = Float64(fma(Float64(cos(x) - cos(y)), Float64(sqrt(2.0) * Float64(fma(sin(x), -0.0625, sin(y)) * sin(x))), 2.0) * Float64(0.3333333333333333 / fma(cos(y), fma(sqrt(5.0), -0.5, 1.5), fma(cos(x), t_0, 1.0))))
	tmp = 0.0
	if (x <= -0.0105)
		tmp = t_1;
	elseif (x <= 0.0048)
		tmp = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * fma(-0.0625, x, sin(y)))), 2.0) / fma(fma(t_0, cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[y], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0105], t$95$1, If[LessEqual[x, 0.0048], N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * x + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0105000000000000007 or 0.00479999999999999958 < x

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 63.1

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

   7. Applied rewrites 63.1%

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

   9. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}} \]

   if -0.0105000000000000007 < x < 0.00479999999999999958

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-rgt-out N/A

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

       4. lower-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-sin.f64 99.0

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

   11. Applied rewrites 99.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0105:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x - \cos y, \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \sin x\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos y, \mathsf{fma}\left(\sqrt{5}, -0.5, 1.5\right), \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ t_1 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_2 := \frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(t\_0, 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{if}\;y \leq -0.0072:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)\right), 2\right)}{\mathsf{fma}\left(t\_0, 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
        (t_1 (fma (sin y) -0.0625 (sin x)))
        (t_2
         (/
          (fma t_1 (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y)))) 2.0)
          (fma t_0 3.0 (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))))
   (if (<= y -0.0072)
     t_2
     (if (<= y 1.2e-6)
       (/
        (fma
         t_1
         (* (sqrt 2.0) (* (+ (cos x) -1.0) (fma -0.0625 (sin x) y)))
         2.0)
        (fma t_0 3.0 (* (* (cos y) 3.0) (- 1.5 (* (sqrt 5.0) 0.5)))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double t_1 = fma(sin(y), -0.0625, sin(x));
	double t_2 = fma(t_1, (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 2.0) / fma(t_0, 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	double tmp;
	if (y <= -0.0072) {
		tmp = t_2;
	} else if (y <= 1.2e-6) {
		tmp = fma(t_1, (sqrt(2.0) * ((cos(x) + -1.0) * fma(-0.0625, sin(x), y))), 2.0) / fma(t_0, 3.0, ((cos(y) * 3.0) * (1.5 - (sqrt(5.0) * 0.5))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	t_1 = fma(sin(y), -0.0625, sin(x))
	t_2 = Float64(fma(t_1, Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / fma(t_0, 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))))
	tmp = 0.0
	if (y <= -0.0072)
		tmp = t_2;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * fma(-0.0625, sin(x), y))), 2.0) / fma(t_0, 3.0, Float64(Float64(cos(y) * 3.0) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0072], t$95$2, If[LessEqual[y, 1.2e-6], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0071999999999999998 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 59.7

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

   11. Applied rewrites 59.7%

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

   if -0.0071999999999999998 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0072:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(t\_0, 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{\mathsf{fma}\left(t\_0, 3, 4.5 + 3 \cdot \left(\sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
        (t_1
         (/
          (fma
           (fma (sin y) -0.0625 (sin x))
           (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y))))
           2.0)
          (fma t_0 3.0 (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))))
   (if (<= y -78.0)
     t_1
     (if (<= y 1.2e-6)
       (/
        (+
         2.0
         (*
          (- (cos x) (cos y))
          (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
        (fma t_0 3.0 (+ 4.5 (* 3.0 (* (sqrt 5.0) -0.5)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double t_1 = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 2.0) / fma(t_0, 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	double tmp;
	if (y <= -78.0) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / fma(t_0, 3.0, (4.5 + (3.0 * (sqrt(5.0) * -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	t_1 = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / fma(t_0, 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))))) / fma(t_0, 3.0, Float64(4.5 + Float64(3.0 * Float64(sqrt(5.0) * -0.5)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$1, If[LessEqual[y, 1.2e-6], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(4.5 + N[(3.0 * N[(N[Sqrt[5.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 60.0

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

   11. Applied rewrites 60.0%

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

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-sin.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. lower-+.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 99.0

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, 4.5 + 3 \cdot \left(\color{blue}{\sqrt{5}} \cdot -0.5\right)\right)} \]

   10. Applied rewrites 99.0%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \color{blue}{4.5 + 3 \cdot \left(\sqrt{5} \cdot -0.5\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, 4.5 + 3 \cdot \left(\sqrt{5} \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (fma (sin y) -0.0625 (sin x))
           (* (sqrt 2.0) (* (sin y) (- 1.0 (cos y))))
           2.0)
          (fma
           (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
           3.0
           (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))))
   (if (<= y -78.0)
     t_0
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (- 3.0 (sqrt 5.0))) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(fma(sin(y), -0.0625, sin(x)), (sqrt(2.0) * (sin(y) * (1.0 - cos(y)))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	double tmp;
	if (y <= -78.0) {
		tmp = t_0;
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), (3.0 - sqrt(5.0))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sqrt(2.0) * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_0;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(3.0 - sqrt(5.0))), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$0, If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-cos.f64 60.0

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

   11. Applied rewrites 60.0%

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

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (fma (sin y) -0.0625 (sin x))
           (* (sin y) (* (sqrt 2.0) (- 1.0 (cos y))))
           2.0)
          (fma
           (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
           3.0
           (* (* (cos y) 3.0) (- 1.5 (* (sqrt 5.0) 0.5)))))))
   (if (<= y -78.0)
     t_0
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (- 3.0 (sqrt 5.0))) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma(fma(sin(y), -0.0625, sin(x)), (sin(y) * (sqrt(2.0) * (1.0 - cos(y)))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((cos(y) * 3.0) * (1.5 - (sqrt(5.0) * 0.5))));
	double tmp;
	if (y <= -78.0) {
		tmp = t_0;
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), (3.0 - sqrt(5.0))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(sin(y), -0.0625, sin(x)), Float64(sin(y) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 3.0) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_0;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(3.0 - sqrt(5.0))), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$0, If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 60.0

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

   9. Applied rewrites 60.0%

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

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sin y \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ t_1 := \sqrt{5} + 3\\ \mathbf{if}\;x \leq -0.00175:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(t\_0, 3, \frac{\cos y \cdot 6}{t\_1}\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(t\_0, 3, \left(\cos y \cdot 3\right) \cdot \frac{2}{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
        (t_1 (+ (sqrt 5.0) 3.0)))
   (if (<= x -0.00175)
     (/
      (fma (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)) 2.0)
      (fma t_0 3.0 (/ (* (cos y) 6.0) t_1)))
     (if (<= x 0.00142)
       (/
        (fma (* (sqrt 2.0) (- 1.0 (cos y))) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma t_0 3.0 (* (* (cos y) 3.0) (/ 2.0 t_1))))
       (/
        (fma
         (- 0.5 (* 0.5 (cos (+ x x))))
         (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
         2.0)
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double t_1 = sqrt(5.0) + 3.0;
	double tmp;
	if (x <= -0.00175) {
		tmp = fma(pow(sin(x), 2.0), (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(t_0, 3.0, ((cos(y) * 6.0) / t_1));
	} else if (x <= 0.00142) {
		tmp = fma((sqrt(2.0) * (1.0 - cos(y))), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(t_0, 3.0, ((cos(y) * 3.0) * (2.0 / t_1)));
	} else {
		tmp = fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	t_1 = Float64(sqrt(5.0) + 3.0)
	tmp = 0.0
	if (x <= -0.00175)
		tmp = Float64(fma((sin(x) ^ 2.0), Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(t_0, 3.0, Float64(Float64(cos(y) * 6.0) / t_1)));
	elseif (x <= 0.00142)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(1.0 - cos(y))), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(t_0, 3.0, Float64(Float64(cos(y) * 3.0) * Float64(2.0 / t_1))));
	else
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[x, -0.00175], N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00142], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] * N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00175000000000000004

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

   11. Applied rewrites 53.7%

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

   if -0.00175000000000000004 < x < 0.00142000000000000004

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. div-sub N/A

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

      7. lift--.f64 N/A

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

      8. div-inv N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. rem-square-sqrt N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 98.3

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

   9. Applied rewrites 98.3%

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

   if 0.00142000000000000004 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00175:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \frac{2}{\sqrt{5} + 3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\\ \mathbf{if}\;x \leq -0.00175:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(t\_0, 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(t\_0, 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)))
   (if (<= x -0.00175)
     (/
      (fma (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)) 2.0)
      (fma t_0 3.0 (/ (* (cos y) 6.0) (+ (sqrt 5.0) 3.0))))
     (if (<= x 0.00142)
       (/
        (fma (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* -0.0625 (sqrt 2.0))) 2.0)
        (fma t_0 3.0 (* (* (cos y) 3.0) (- 1.5 (* (sqrt 5.0) 0.5)))))
       (/
        (fma
         (- 0.5 (* 0.5 (cos (+ x x))))
         (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
         2.0)
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0);
	double tmp;
	if (x <= -0.00175) {
		tmp = fma(pow(sin(x), 2.0), (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(t_0, 3.0, ((cos(y) * 6.0) / (sqrt(5.0) + 3.0)));
	} else if (x <= 0.00142) {
		tmp = fma(pow(sin(y), 2.0), ((1.0 - cos(y)) * (-0.0625 * sqrt(2.0))), 2.0) / fma(t_0, 3.0, ((cos(y) * 3.0) * (1.5 - (sqrt(5.0) * 0.5))));
	} else {
		tmp = fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)
	tmp = 0.0
	if (x <= -0.00175)
		tmp = Float64(fma((sin(x) ^ 2.0), Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(t_0, 3.0, Float64(Float64(cos(y) * 6.0) / Float64(sqrt(5.0) + 3.0))));
	elseif (x <= 0.00142)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * sqrt(2.0))), 2.0) / fma(t_0, 3.0, Float64(Float64(cos(y) * 3.0) * Float64(1.5 - Float64(sqrt(5.0) * 0.5)))));
	else
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.00175], N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00142], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$0 * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 3.0), $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00175000000000000004

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

   11. Applied rewrites 53.7%

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

   if -0.00175000000000000004 < x < 0.00142000000000000004

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 0.00142000000000000004 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 66.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00175:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(\cos y \cdot 3\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + 3\\ \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{t\_0}\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{t\_0}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) 3.0)))
   (if (<= x -0.00165)
     (/
      (fma (pow (sin x) 2.0) (* (sqrt 2.0) (fma -0.0625 (cos x) 0.0625)) 2.0)
      (fma
       (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0)
       3.0
       (/ (* (cos y) 6.0) t_0)))
     (if (<= x 0.00142)
       (/
        (fma (* (sqrt 2.0) (- 1.0 (cos y))) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma 6.0 (/ (cos y) t_0) (fma (sqrt 5.0) 1.5 1.5)))
       (/
        (fma
         (- 0.5 (* 0.5 (cos (+ x x))))
         (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
         2.0)
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + 3.0;
	double tmp;
	if (x <= -0.00165) {
		tmp = fma(pow(sin(x), 2.0), (sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, ((cos(y) * 6.0) / t_0));
	} else if (x <= 0.00142) {
		tmp = fma((sqrt(2.0) * (1.0 - cos(y))), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(6.0, (cos(y) / t_0), fma(sqrt(5.0), 1.5, 1.5));
	} else {
		tmp = fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + 3.0)
	tmp = 0.0
	if (x <= -0.00165)
		tmp = Float64(fma((sin(x) ^ 2.0), Float64(sqrt(2.0) * fma(-0.0625, cos(x), 0.0625)), 2.0) / fma(fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0), 3.0, Float64(Float64(cos(y) * 6.0) / t_0)));
	elseif (x <= 0.00142)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(1.0 - cos(y))), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(6.0, Float64(cos(y) / t_0), fma(sqrt(5.0), 1.5, 1.5)));
	else
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[x, -0.00165], N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision] * 3.0 + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00142], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00165

   1. Initial program 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

   6. Applied rewrites 99.1%

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

   8. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \color{blue}{\frac{\cos y \cdot 6}{3 + \sqrt{5}}}\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       6. sub-neg N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \color{blue}{\left(\cos x + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       7. metadata-eval N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       8. distribute-lft-in N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \cos x + \frac{-1}{16} \cdot -1\right)}\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \cos x + \color{blue}{\frac{1}{16}}\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

       10. +-commutative N/A

           \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{16} + \frac{-1}{16} \cdot \cos x\right)}\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

   11. Applied rewrites 53.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

   if -0.00165 < x < 0.00142000000000000004

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3}{2}} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      6. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{3 - \sqrt{5}}}{2}\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right)\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}}\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{2}}{3 + \sqrt{5}}\right)} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{2}{\color{blue}{3 + \sqrt{5}}}\right)} \]

   6. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), {\sin y}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{3 + \sqrt{5}}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}} \]

   if 0.00142000000000000004 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \frac{\cos y \cdot 6}{\sqrt{5} + 3}\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, t\_1, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \cos y \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), t\_1, 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_0}{2}\right) + \cos y \cdot \frac{t\_2}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.00165)
     (/
      (fma (pow (sin x) 2.0) t_1 2.0)
      (fma 1.5 (fma (cos x) t_0 (* (cos y) t_2)) 3.0))
     (if (<= x 0.00142)
       (/
        (fma (* (sqrt 2.0) (- 1.0 (cos y))) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma 6.0 (/ (cos y) (+ (sqrt 5.0) 3.0)) (fma (sqrt 5.0) 1.5 1.5)))
       (/
        (fma (- 0.5 (* 0.5 (cos (+ x x)))) t_1 2.0)
        (*
         3.0
         (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_2 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sqrt(2.0) * fma(cos(x), -0.0625, 0.0625);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.00165) {
		tmp = fma(pow(sin(x), 2.0), t_1, 2.0) / fma(1.5, fma(cos(x), t_0, (cos(y) * t_2)), 3.0);
	} else if (x <= 0.00142) {
		tmp = fma((sqrt(2.0) * (1.0 - cos(y))), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(6.0, (cos(y) / (sqrt(5.0) + 3.0)), fma(sqrt(5.0), 1.5, 1.5));
	} else {
		tmp = fma((0.5 - (0.5 * cos((x + x)))), t_1, 2.0) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_2 / 2.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.00165)
		tmp = Float64(fma((sin(x) ^ 2.0), t_1, 2.0) / fma(1.5, fma(cos(x), t_0, Float64(cos(y) * t_2)), 3.0));
	elseif (x <= 0.00142)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(1.0 - cos(y))), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(6.0, Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), fma(sqrt(5.0), 1.5, 1.5)));
	else
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), t_1, 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_2 / 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00165], N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00142], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00165

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 53.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 52.3%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

       3. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   11. Applied rewrites 53.7%

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]

   if -0.00165 < x < 0.00142000000000000004

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3}{2}} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      6. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{3 - \sqrt{5}}}{2}\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right)\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}}\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{2}}{3 + \sqrt{5}}\right)} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{2}{\color{blue}{3 + \sqrt{5}}}\right)} \]

   6. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), {\sin y}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{3 + \sqrt{5}}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}} \]

   if 0.00142000000000000004 < x

   1. Initial program 99.2%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 66.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (- 0.5 (* 0.5 (cos (+ x x))))
           (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
           2.0)
          (*
           3.0
           (+
            (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
            (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))
   (if (<= x -0.00165)
     t_0
     (if (<= x 0.00142)
       (/
        (fma (* (sqrt 2.0) (- 1.0 (cos y))) (* -0.0625 (pow (sin y) 2.0)) 2.0)
        (fma 6.0 (/ (cos y) (+ (sqrt 5.0) 3.0)) (fma (sqrt 5.0) 1.5 1.5)))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	double tmp;
	if (x <= -0.00165) {
		tmp = t_0;
	} else if (x <= 0.00142) {
		tmp = fma((sqrt(2.0) * (1.0 - cos(y))), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(6.0, (cos(y) / (sqrt(5.0) + 3.0)), fma(sqrt(5.0), 1.5, 1.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))))
	tmp = 0.0
	if (x <= -0.00165)
		tmp = t_0;
	elseif (x <= 0.00142)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64(1.0 - cos(y))), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(6.0, Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), fma(sqrt(5.0), 1.5, 1.5)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00165], t$95$0, If[LessEqual[x, 0.00142], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00165 or 0.00142000000000000004 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 60.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -0.00165 < x < 0.00142000000000000004

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3}{2}} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      6. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{3 - \sqrt{5}}}{2}\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right)\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}}\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{2}}{3 + \sqrt{5}}\right)} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

      20. lower-+.f64 99.8

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{2}{\color{blue}{3 + \sqrt{5}}}\right)} \]

   6. Applied rewrites 99.8%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), {\sin y}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{3 + \sqrt{5}}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00165:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00142:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)\\ t_2 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(t\_0, \cos y \cdot 0.5, 1\right)\right)\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{1}{\frac{3 \cdot t\_2}{t\_1}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_2}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          (* (sqrt 2.0) (* -0.0625 (- 1.0 (cos y))))
          (- 0.5 (* 0.5 (cos (+ y y))))
          2.0))
        (t_2
         (fma
          (cos x)
          (fma (sqrt 5.0) 0.5 -0.5)
          (fma t_0 (* (cos y) 0.5) 1.0))))
   (if (<= y -78.0)
     (/ 1.0 (/ (* 3.0 t_2) t_1))
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       (/ (/ t_1 t_2) 3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))), (0.5 - (0.5 * cos((y + y)))), 2.0);
	double t_2 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, (cos(y) * 0.5), 1.0));
	double tmp;
	if (y <= -78.0) {
		tmp = 1.0 / ((3.0 * t_2) / t_1);
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = (t_1 / t_2) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64(sqrt(2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))), Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), 2.0)
	t_2 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, Float64(cos(y) * 0.5), 1.0))
	tmp = 0.0
	if (y <= -78.0)
		tmp = Float64(1.0 / Float64(Float64(3.0 * t_2) / t_1));
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = Float64(Float64(t_1 / t_2) / 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[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], N[(1.0 / N[(N[(3.0 * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -78

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      16. lower-sqrt.f64 54.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 54.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   9. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)}{\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)}}} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]

   if 1.1999999999999999e-6 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. 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 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      16. lower-sqrt.f64 63.1

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 63.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   9. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)}}{3}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)\\ t_2 := \mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(t\_0, \cos y \cdot 0.5, 1\right)\right)\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_1 \cdot \frac{-1}{t\_2 \cdot -3}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_2}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (fma
          (* (sqrt 2.0) (* -0.0625 (- 1.0 (cos y))))
          (- 0.5 (* 0.5 (cos (+ y y))))
          2.0))
        (t_2
         (fma
          (cos x)
          (fma (sqrt 5.0) 0.5 -0.5)
          (fma t_0 (* (cos y) 0.5) 1.0))))
   (if (<= y -78.0)
     (* t_1 (/ -1.0 (* t_2 -3.0)))
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       (/ (/ t_1 t_2) 3.0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))), (0.5 - (0.5 * cos((y + y)))), 2.0);
	double t_2 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, (cos(y) * 0.5), 1.0));
	double tmp;
	if (y <= -78.0) {
		tmp = t_1 * (-1.0 / (t_2 * -3.0));
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = (t_1 / t_2) / 3.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = fma(Float64(sqrt(2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))), Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), 2.0)
	t_2 = fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, Float64(cos(y) * 0.5), 1.0))
	tmp = 0.0
	if (y <= -78.0)
		tmp = Float64(t_1 * Float64(-1.0 / Float64(t_2 * -3.0)));
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = Float64(Float64(t_1 / t_2) / 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[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], N[(t$95$1 * N[(-1.0 / N[(t$95$2 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$2), $MachinePrecision] / 3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -78

   1. Initial program 98.9%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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 98.9

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      16. lower-sqrt.f64 54.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 54.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   9. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\left(-\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right) \cdot -3}} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]

   if 1.1999999999999999e-6 < y

   1. Initial program 99.3%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. 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 99.4

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, \color{blue}{1 + \frac{3 - \sqrt{5}}{2} \cdot \cos y}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      16. lower-sqrt.f64 63.1

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 63.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   9. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)}}{3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right) \cdot \frac{-1}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right) \cdot -3}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(t\_0, \cos y \cdot 0.5, 1\right)\right)}}{3}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (/
           (fma
            (* (sqrt 2.0) (* -0.0625 (- 1.0 (cos y))))
            (- 0.5 (* 0.5 (cos (+ y y))))
            2.0)
           (fma
            (cos x)
            (fma (sqrt 5.0) 0.5 -0.5)
            (fma t_0 (* (cos y) 0.5) 1.0)))
          3.0)))
   (if (<= y -78.0)
     t_1
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = (fma((sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))), (0.5 - (0.5 * cos((y + y)))), 2.0) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, (cos(y) * 0.5), 1.0))) / 3.0;
	double tmp;
	if (y <= -78.0) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(fma(Float64(sqrt(2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))), Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), 2.0) / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, Float64(cos(y) * 0.5), 1.0))) / 3.0)
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$1, If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      16. lower-sqrt.f64 58.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 58.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   9. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)}}{3}} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(t\_0, \cos y \cdot 0.5, 1\right)\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          (fma
           (* (sqrt 2.0) (* -0.0625 (- 1.0 (cos y))))
           (- 0.5 (* 0.5 (cos (+ y y))))
           2.0)
          (/
           0.3333333333333333
           (fma
            (cos x)
            (fma (sqrt 5.0) 0.5 -0.5)
            (fma t_0 (* (cos y) 0.5) 1.0))))))
   (if (<= y -78.0)
     t_1
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))), (0.5 - (0.5 * cos((y + y)))), 2.0) * (0.3333333333333333 / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, (cos(y) * 0.5), 1.0)));
	double tmp;
	if (y <= -78.0) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))), Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), 2.0) * Float64(0.3333333333333333 / fma(cos(x), fma(sqrt(5.0), 0.5, -0.5), fma(t_0, Float64(cos(y) * 0.5), 1.0))))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(0.3333333333333333 / N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] + N[(t$95$0 * N[(N[Cos[y], $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$1, If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\sin y}^{2}}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right) \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)} \cdot \sqrt{2}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}, 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      16. lower-sqrt.f64 58.6

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \color{blue}{\sqrt{2}}\right), 2\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 58.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \sqrt{2}\right), 2\right)}}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   9. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)} \cdot \mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right)} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.5 - 0.5 \cdot \cos \left(y + y\right), 2\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \mathsf{fma}\left(3 - \sqrt{5}, \cos y \cdot 0.5, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (sqrt 2.0) (- 1.0 (cos y)))
           (* -0.0625 (pow (sin y) 2.0))
           2.0)
          (fma 6.0 (/ (cos y) (+ (sqrt 5.0) 3.0)) (fma (sqrt 5.0) 1.5 1.5)))))
   (if (<= y -78.0)
     t_0
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (- 3.0 (sqrt 5.0))) 1.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((sqrt(2.0) * (1.0 - cos(y))), (-0.0625 * pow(sin(y), 2.0)), 2.0) / fma(6.0, (cos(y) / (sqrt(5.0) + 3.0)), fma(sqrt(5.0), 1.5, 1.5));
	double tmp;
	if (y <= -78.0) {
		tmp = t_0;
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), (3.0 - sqrt(5.0))), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(sqrt(2.0) * Float64(1.0 - cos(y))), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / fma(6.0, Float64(cos(y) / Float64(sqrt(5.0) + 3.0)), fma(sqrt(5.0), 1.5, 1.5)))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_0;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(3.0 - sqrt(5.0))), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * 1.5 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$0, If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3}{2}} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\sqrt{5} \cdot \frac{1}{2}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      5. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \color{blue}{\frac{\sqrt{5}}{2}}\right)\right)} \]

      6. div-sub N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{3 - \sqrt{5}}{2}}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{3 - \sqrt{5}}}{2}\right)} \]

      8. div-inv N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\left(\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}\right)}\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right)\right)} \]

      10. flip-- N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}} \cdot \frac{1}{2}\right)\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}} \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      12. associate-*l/ N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{\left(3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}}\right)} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{\sqrt{5}} \cdot \sqrt{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \sqrt{5} \cdot \color{blue}{\sqrt{5}}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(3 \cdot 3 - \color{blue}{5}\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\left(\color{blue}{9} - 5\right) \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      17. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{4} \cdot \frac{1}{2}}{3 + \sqrt{5}}\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{\color{blue}{2}}{3 + \sqrt{5}}\right)} \]

      19. lower-/.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

      20. lower-+.f64 99.3

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \frac{2}{\color{blue}{3 + \sqrt{5}}}\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \color{blue}{\frac{2}{3 + \sqrt{5}}}\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}}} \]

   9. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), {\sin y}^{2} \cdot -0.0625, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{3 + \sqrt{5}}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{\mathsf{fma}\left(6, \frac{\cos y}{\sqrt{5} + 3}, \mathsf{fma}\left(\sqrt{5}, 1.5, 1.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 0.5\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (sqrt 2.0) (- 1.0 (cos y)))
           (* -0.0625 (pow (sin y) 2.0))
           2.0)
          (* 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) 0.5)))))
   (if (<= y -78.0)
     t_1
     (if (<= y 1.2e-6)
       (/
        (fma
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma -0.0625 (cos x) 0.0625)))
         0.3333333333333333
         0.6666666666666666)
        (fma 0.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (1.0 - cos(y))), (-0.0625 * pow(sin(y), 2.0)), 2.0) / (3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5));
	double tmp;
	if (y <= -78.0) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = fma((sqrt(2.0) * (pow(sin(x), 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * Float64(1.0 - cos(y))), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / Float64(3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5)))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(-0.0625, cos(x), 0.0625))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$1, If[LessEqual[y, 1.2e-6], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\cos y \cdot \left(\frac{3}{2} - \frac{1}{2} \cdot \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), {\sin y}^{2} \cdot -0.0625, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Applied rewrites 99.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right)\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 0.5\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (sqrt 2.0) (- 1.0 (cos y)))
           (* -0.0625 (pow (sin y) 2.0))
           2.0)
          (* 3.0 (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) 0.5)))))
   (if (<= y -78.0)
     t_1
     (if (<= y 1.2e-6)
       (*
        (fma
         (- 0.5 (* 0.5 (cos (+ x x))))
         (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
         2.0)
        (/ 1.0 (fma 1.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma((sqrt(2.0) * (1.0 - cos(y))), (-0.0625 * pow(sin(y), 2.0)), 2.0) / (3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5));
	double tmp;
	if (y <= -78.0) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * (1.0 / fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(sqrt(2.0) * Float64(1.0 - cos(y))), Float64(-0.0625 * (sin(y) ^ 2.0)), 2.0) / Float64(3.0 * fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5)))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * Float64(1.0 / fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$1, If[LessEqual[y, 1.2e-6], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / 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]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) \cdot 3 + \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x, 3, \left(\frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3\right)}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)} \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)} + 2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(\frac{3}{2} - \sqrt{5} \cdot \frac{1}{2}\right)\right)} \]

   6. Applied rewrites 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin y, -0.0625, \sin x\right), \sqrt{2} \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right), 3, \left(3 \cdot \cos y\right) \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\cos y \cdot \left(\frac{3}{2} - \frac{1}{2} \cdot \sqrt{5}\right)\right) + 3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{5}\right)}} \]

   9. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), {\sin y}^{2} \cdot -0.0625, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   10. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(1 - \cos y\right), -0.0625 \cdot {\sin y}^{2}, 2\right)}{3 \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5}\right), 0.5\right)}\\ \mathbf{if}\;y \leq -78:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           (* (* (sqrt 2.0) (- 1.0 (cos y))) (* -0.0625 (pow (sin y) 2.0)))
           0.3333333333333333
           0.6666666666666666)
          (fma 0.5 (fma (cos y) t_0 (sqrt 5.0)) 0.5))))
   (if (<= y -78.0)
     t_1
     (if (<= y 1.2e-6)
       (*
        (fma
         (- 0.5 (* 0.5 (cos (+ x x))))
         (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
         2.0)
        (/ 1.0 (fma 1.5 (fma (cos x) (+ (sqrt 5.0) -1.0) t_0) 3.0)))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(((sqrt(2.0) * (1.0 - cos(y))) * (-0.0625 * pow(sin(y), 2.0))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5);
	double tmp;
	if (y <= -78.0) {
		tmp = t_1;
	} else if (y <= 1.2e-6) {
		tmp = fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * (1.0 / fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), t_0), 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(fma(Float64(Float64(sqrt(2.0) * Float64(1.0 - cos(y))) * Float64(-0.0625 * (sin(y) ^ 2.0))), 0.3333333333333333, 0.6666666666666666) / fma(0.5, fma(cos(y), t_0, sqrt(5.0)), 0.5))
	tmp = 0.0
	if (y <= -78.0)
		tmp = t_1;
	elseif (y <= 1.2e-6)
		tmp = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * Float64(1.0 / fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), t_0), 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -78.0], t$95$1, If[LessEqual[y, 1.2e-6], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / 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]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -78 or 1.1999999999999999e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

      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%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left({\sin y}^{2} \cdot \sqrt{2}\right) \cdot \frac{-1}{16}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{2 + \color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \frac{-1}{16}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left({\sin y}^{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2 + \color{blue}{\left({\sin y}^{2} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{{\sin y}^{2}} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{2 + \left({\color{blue}{\sin y}}^{2} \cdot \left(\frac{-1}{16} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left({\sin y}^{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(\frac{1}{2} \cdot \cos y\right)\right)} \]

      8. lower-sqrt.f64 58.8

         \[\leadsto \frac{2 + \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   7. Applied rewrites 58.8%

      \[\leadsto \frac{2 + \color{blue}{\left({\sin y}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1 + \left(3 - \sqrt{5}\right) \cdot \left(0.5 \cdot \cos y\right)\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

   10. Applied rewrites 57.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}} \]

   if -78 < y < 1.1999999999999999e-6

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

   10. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right), 0.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 60.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (fma
   (- 0.5 (* 0.5 (cos (+ x x))))
   (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
   2.0)
  (/ 1.0 (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((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * (1.0 / 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(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) * Float64(1.0 / 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[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(1.0 / 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]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

10. Applied rewrites 63.2%

    \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \cdot \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)} \]

11. Final simplification 63.2%

    \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
12. Add Preprocessing

Alternative 32: 60.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- 0.5 (* 0.5 (cos (+ x x))))
   (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
   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((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 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(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 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[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $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 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 63.2%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]
11. Add Preprocessing

Alternative 33: 45.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   1.5
   (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) (- 3.0 (sqrt 5.0))))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * (3.0 - sqrt(5.0)))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * Float64(3.0 - sqrt(5.0)))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{neg}\left(\frac{-3}{2}\right)\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    7. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\mathsf{neg}\left(\frac{-3}{2}\right)\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    8. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{-3}{2}\right), \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 48.7%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]
15. Add Preprocessing

Alternative 34: 43.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \cos x \cdot \sqrt{5} + \left(\left(3 - \sqrt{5}\right) - \cos x\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma 1.5 (+ (* (cos x) (sqrt 5.0)) (- (- 3.0 (sqrt 5.0)) (cos x))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, ((cos(x) * sqrt(5.0)) + ((3.0 - sqrt(5.0)) - cos(x))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, Float64(Float64(cos(x) * sqrt(5.0)) + Float64(Float64(3.0 - sqrt(5.0)) - cos(x))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
12. Step-by-step derivation

13. Applied rewrites 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \cos x \cdot \sqrt{5} + \color{blue}{\left(\left(-\cos x\right) + \left(3 - \sqrt{5}\right)\right)}, 3\right)} \]

14. Final simplification 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \cos x \cdot \sqrt{5} + \left(\left(3 - \sqrt{5}\right) - \cos x\right), 3\right)} \]
15. Add Preprocessing

Alternative 35: 43.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (fma (+ (sqrt 5.0) -1.0) (cos x) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma((sqrt(5.0) + -1.0), cos(x), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(Float64(sqrt(5.0) + -1.0), cos(x), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
12. Step-by-step derivation

13. Applied rewrites 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} + -1, \color{blue}{\cos x}, 3 - \sqrt{5}\right), 3\right)} \]
14. Add Preprocessing

Alternative 36: 43.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \end{array} \]

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, Float64(fma(cos(x), Float64(sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
12. Add Preprocessing

Alternative 37: 40.5% accurate, 52.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, 2, 3\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 2.0 (fma 1.5 2.0 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, 2.0, 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, 2.0, 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * 2.0 + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 65.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

8. Applied rewrites 63.1%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 46.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, 2, 3\right)} \]
13. Step-by-step derivation

14. Applied rewrites 44.6%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, 2, 3\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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))))))