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

Percentage Accurate: 99.3% → 99.3%

Time: 19.6s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (*
     (* (sqrt 2.0) (fma -0.0625 (sin y) (sin x)))
     (- (sin y) (/ (sin x) 16.0)))
    (- (cos x) (cos y))))
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5)))
    (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))

C

double code(double x, double y) {
	return (2.0 + (((sqrt(2.0) * fma(-0.0625, sin(y), sin(x))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * fma(-0.0625, sin(y), sin(x))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-sin.f64 99.3

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

5. Simplified 99.3%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-sqrt.f64 99.3

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-sin.f64 99.3

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

5. Simplified 99.3%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-sqrt.f64 99.3

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

8. Simplified 99.3%

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

9. Taylor expanded in y around inf

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

    1. metadata-eval N/A

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

    2. cancel-sign-sub-inv N/A

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

    3. lower-*.f64 N/A

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

    4. lower-sqrt.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. sub-neg N/A

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

    8. +-commutative N/A

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

    9. *-commutative N/A

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

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

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

    11. metadata-eval N/A

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

    12. lower-fma.f64 N/A

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

    13. lower-sin.f64 N/A

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

    14. lower-sin.f64 N/A

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

    15. cancel-sign-sub-inv N/A

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

    16. metadata-eval N/A

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 3: 81.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_1
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
          t_0)))
   (if (<= x -0.7)
     t_1
     (if (<= x 0.37)
       (/
        (+
         2.0
         (*
          (*
           (sqrt 2.0)
           (* (fma -0.0625 (sin y) (sin x)) (fma (sin x) -0.0625 (sin y))))
          (fma
           (* x x)
           (fma
            (* x x)
            (fma (* x x) -0.001388888888888889 0.041666666666666664)
            -0.5)
           (- 1.0 (cos y)))))
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_1 = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / t_0;
	double tmp;
	if (x <= -0.7) {
		tmp = t_1;
	} else if (x <= 0.37) {
		tmp = (2.0 + ((sqrt(2.0) * (fma(-0.0625, sin(y), sin(x)) * fma(sin(x), -0.0625, sin(y)))) * fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), (1.0 - cos(y))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_1 = 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))))) / t_0)
	tmp = 0.0
	if (x <= -0.7)
		tmp = t_1;
	elseif (x <= 0.37)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(fma(-0.0625, sin(y), sin(x)) * fma(sin(x), -0.0625, sin(y)))) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), Float64(1.0 - cos(y))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.7], t$95$1, If[LessEqual[x, 0.37], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.37 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.9

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

   8. Simplified 98.9%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-sqrt.f64 59.9

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

   11. Simplified 59.9%

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

   if -0.69999999999999996 < x < 0.37

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 99.6

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. sub-neg N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-sin.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. cancel-sign-sub-inv N/A

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

       16. metadata-eval N/A

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

   11. Simplified 99.6%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. lower-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower--.f64 N/A

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

       17. lower-cos.f64 99.4

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

   14. Simplified 99.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.37:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.32:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), x, 1 - \cos y\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_1
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
          t_0)))
   (if (<= x -0.32)
     t_1
     (if (<= x 0.32)
       (/
        (+
         2.0
         (*
          (*
           (sqrt 2.0)
           (* (fma -0.0625 (sin y) (sin x)) (fma (sin x) -0.0625 (sin y))))
          (fma
           (* x (fma x (* x 0.041666666666666664) -0.5))
           x
           (- 1.0 (cos y)))))
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_1 = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / t_0;
	double tmp;
	if (x <= -0.32) {
		tmp = t_1;
	} else if (x <= 0.32) {
		tmp = (2.0 + ((sqrt(2.0) * (fma(-0.0625, sin(y), sin(x)) * fma(sin(x), -0.0625, sin(y)))) * fma((x * fma(x, (x * 0.041666666666666664), -0.5)), x, (1.0 - cos(y))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_1 = 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))))) / t_0)
	tmp = 0.0
	if (x <= -0.32)
		tmp = t_1;
	elseif (x <= 0.32)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(fma(-0.0625, sin(y), sin(x)) * fma(sin(x), -0.0625, sin(y)))) * fma(Float64(x * fma(x, Float64(x * 0.041666666666666664), -0.5)), x, Float64(1.0 - cos(y))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.32], t$95$1, If[LessEqual[x, 0.32], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.320000000000000007 or 0.320000000000000007 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.9

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

   8. Simplified 98.9%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-sqrt.f64 59.9

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

   11. Simplified 59.9%

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

   if -0.320000000000000007 < x < 0.320000000000000007

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 99.6

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. sub-neg N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-sin.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. cancel-sign-sub-inv N/A

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

       16. metadata-eval N/A

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

   11. Simplified 99.6%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

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

       9. *-commutative N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower--.f64 N/A

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

       16. lower-cos.f64 99.0

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

   14. Simplified 99.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.32:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, -0.5\right), x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + t\_0\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.092:\\ \;\;\;\;\frac{2 + \left(t\_1 \cdot \mathsf{fma}\left(x, \left(x \cdot \left(\sqrt{2} \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(t\_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2
         (/
          (+ 2.0 (* (- (cos x) (cos y)) (* t_1 (* (sqrt 2.0) (sin x)))))
          (* 3.0 (+ (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5))) t_0)))))
   (if (<= x -0.31)
     t_2
     (if (<= x 0.092)
       (/
        (+
         2.0
         (*
          (*
           t_1
           (fma
            x
            (*
             (* x (* (sqrt 2.0) x))
             (fma (* x x) 0.008333333333333333 -0.16666666666666666))
            (* (sqrt 2.0) (fma -0.0625 (sin y) x))))
          (fma -0.5 (* x x) (- 1.0 (cos y)))))
        (* 3.0 (+ t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = (2.0 + ((cos(x) - cos(y)) * (t_1 * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0));
	double tmp;
	if (x <= -0.31) {
		tmp = t_2;
	} else if (x <= 0.092) {
		tmp = (2.0 + ((t_1 * fma(x, ((x * (sqrt(2.0) * x)) * fma((x * x), 0.008333333333333333, -0.16666666666666666)), (sqrt(2.0) * fma(-0.0625, sin(y), x)))) * fma(-0.5, (x * x), (1.0 - cos(y))))) / (3.0 * (t_0 + (1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_1 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0)))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_2;
	elseif (x <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * fma(x, Float64(Float64(x * Float64(sqrt(2.0) * x)) * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666)), Float64(sqrt(2.0) * fma(-0.0625, sin(y), x)))) * fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$2, If[LessEqual[x, 0.092], N[(N[(2.0 + N[(N[(t$95$1 * N[(x * N[(N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.091999999999999998 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.9

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.9

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

   8. Simplified 98.9%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-sqrt.f64 59.9

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

   11. Simplified 59.9%

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

   if -0.309999999999999998 < x < 0.091999999999999998

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \sqrt{2} + x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right) + x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right) + \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot x + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)} + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right), \frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 98.7%

      \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, \left(x \cdot \left(x \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.092:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \left(x \cdot \left(\sqrt{2} \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + t\_0\right)\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{t\_1}\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \left(x \cdot \left(\sqrt{2} \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(t\_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x + -1\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_1 (* 3.0 (+ (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5))) t_0))))
   (if (<= x -0.31)
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0)))))
      t_1)
     (if (<= x 0.1)
       (/
        (+
         2.0
         (*
          (*
           (- (sin y) (/ (sin x) 16.0))
           (fma
            x
            (*
             (* x (* (sqrt 2.0) x))
             (fma (* x x) 0.008333333333333333 -0.16666666666666666))
            (* (sqrt 2.0) (fma -0.0625 (sin y) x))))
          (fma -0.5 (* x x) (- 1.0 (cos y)))))
        (* 3.0 (+ t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))))
       (/
        (+
         2.0
         (*
          (*
           (sqrt 2.0)
           (* (fma -0.0625 (sin y) (sin x)) (fma (sin x) -0.0625 (sin y))))
          (+ (cos x) -1.0)))
        t_1)))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0);
	double tmp;
	if (x <= -0.31) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))))) / t_1;
	} else if (x <= 0.1) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * fma(x, ((x * (sqrt(2.0) * x)) * fma((x * x), 0.008333333333333333, -0.16666666666666666)), (sqrt(2.0) * fma(-0.0625, sin(y), x)))) * fma(-0.5, (x * x), (1.0 - cos(y))))) / (3.0 * (t_0 + (1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0)))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (fma(-0.0625, sin(y), sin(x)) * fma(sin(x), -0.0625, sin(y)))) * (cos(x) + -1.0))) / t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0))
	tmp = 0.0
	if (x <= -0.31)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / t_1);
	elseif (x <= 0.1)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * fma(x, Float64(Float64(x * Float64(sqrt(2.0) * x)) * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666)), Float64(sqrt(2.0) * fma(-0.0625, sin(y), x)))) * fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(fma(-0.0625, sin(y), sin(x)) * fma(sin(x), -0.0625, sin(y)))) * Float64(cos(x) + -1.0))) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 0.1], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.309999999999999998

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 55.0

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

   8. Simplified 55.0%

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

   if -0.309999999999999998 < x < 0.10000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \sqrt{2} + x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right) + x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right) + \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot x + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)} + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right), \frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 98.7%

      \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, \left(x \cdot \left(x \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if 0.10000000000000001 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.8

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 98.8

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

   8. Simplified 98.8%

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

   9. Taylor expanded in y around inf

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

       1. metadata-eval N/A

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

       2. cancel-sign-sub-inv N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. sub-neg N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

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

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-sin.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. cancel-sign-sub-inv N/A

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

       16. metadata-eval N/A

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

   11. Simplified 98.8%

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

   12. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. lower-+.f64 N/A

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

       4. lower-cos.f64 58.2

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

   14. Simplified 58.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \left(x \cdot \left(\sqrt{2} \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right) \cdot \mathsf{fma}\left(\sin x, -0.0625, \sin y\right)\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + t\_0\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \left(x \cdot \left(\sqrt{2} \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(t\_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_1
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0)))))
          (* 3.0 (+ (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5))) t_0)))))
   (if (<= x -0.31)
     t_1
     (if (<= x 0.1)
       (/
        (+
         2.0
         (*
          (*
           (- (sin y) (/ (sin x) 16.0))
           (fma
            x
            (*
             (* x (* (sqrt 2.0) x))
             (fma (* x x) 0.008333333333333333 -0.16666666666666666))
            (* (sqrt 2.0) (fma -0.0625 (sin y) x))))
          (fma -0.5 (* x x) (- 1.0 (cos y)))))
        (* 3.0 (+ t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))))) / (3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0));
	double tmp;
	if (x <= -0.31) {
		tmp = t_1;
	} else if (x <= 0.1) {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * fma(x, ((x * (sqrt(2.0) * x)) * fma((x * x), 0.008333333333333333, -0.16666666666666666)), (sqrt(2.0) * fma(-0.0625, sin(y), x)))) * fma(-0.5, (x * x), (1.0 - cos(y))))) / (3.0 * (t_0 + (1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0)))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_1;
	elseif (x <= 0.1)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * fma(x, Float64(Float64(x * Float64(sqrt(2.0) * x)) * fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666)), Float64(sqrt(2.0) * fma(-0.0625, sin(y), x)))) * fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$1, If[LessEqual[x, 0.1], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(x * N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.10000000000000001 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 56.5

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

   8. Simplified 56.5%

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

   if -0.309999999999999998 < x < 0.10000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \sqrt{2} + x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right) + x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right) + \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot x + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right)\right)} + \left(\frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{-1}{6} \cdot \sqrt{2} + \frac{1}{120} \cdot \left({x}^{2} \cdot \sqrt{2}\right)\right), \frac{-1}{16} \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 98.7%

      \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, \left(x \cdot \left(x \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \left(x \cdot \left(\sqrt{2} \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), \sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + t\_0\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{3 \cdot \left(t\_0 + \left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.020833333333333332, -0.25\right), 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_1
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0)))))
          (* 3.0 (+ (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5))) t_0)))))
   (if (<= x -0.31)
     t_1
     (if (<= x 0.1)
       (/
        (+
         2.0
         (*
          (fma -0.5 (* x x) (- 1.0 (cos y)))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))))
        (*
         3.0
         (+
          t_0
          (+
           1.0
           (*
            (+ (sqrt 5.0) -1.0)
            (fma (* x x) (fma (* x x) 0.020833333333333332 -0.25) 0.5))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))))) / (3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0));
	double tmp;
	if (x <= -0.31) {
		tmp = t_1;
	} else if (x <= 0.1) {
		tmp = (2.0 + (fma(-0.5, (x * x), (1.0 - cos(y))) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0)))))) / (3.0 * (t_0 + (1.0 + ((sqrt(5.0) + -1.0) * fma((x * x), fma((x * x), 0.020833333333333332, -0.25), 0.5)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0)))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_1;
	elseif (x <= 0.1)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0)))))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) * fma(Float64(x * x), fma(Float64(x * x), 0.020833333333333332, -0.25), 0.5))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$1, If[LessEqual[x, 0.1], N[(N[(2.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.10000000000000001 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 56.5

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

   8. Simplified 56.5%

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

   if -0.309999999999999998 < x < 0.10000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt{5} - 1\right) + {x}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{48} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{4} \cdot \left(\sqrt{5} - 1\right) + \frac{1}{48} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 98.7%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.020833333333333332, -0.25\right), 0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.1:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.020833333333333332, -0.25\right), 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + t\_0\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.092:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\right)\right)\right)}{3 \cdot \left(t\_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_1
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0)))))
          (* 3.0 (+ (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5))) t_0)))))
   (if (<= x -0.31)
     t_1
     (if (<= x 0.092)
       (/
        (+
         2.0
         (*
          (fma -0.5 (* x x) (- 1.0 (cos y)))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (*
            (sqrt 2.0)
            (fma
             -0.0625
             (sin y)
             (fma x (* (* x x) -0.16666666666666666) x))))))
        (* 3.0 (+ t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))))) / (3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0));
	double tmp;
	if (x <= -0.31) {
		tmp = t_1;
	} else if (x <= 0.092) {
		tmp = (2.0 + (fma(-0.5, (x * x), (1.0 - cos(y))) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(-0.0625, sin(y), fma(x, ((x * x) * -0.16666666666666666), x)))))) / (3.0 * (t_0 + (1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0)))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_1;
	elseif (x <= 0.092)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), fma(x, Float64(Float64(x * x) * -0.16666666666666666), x)))))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$1, If[LessEqual[x, 0.092], N[(N[(2.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.091999999999999998 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 56.5

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

   8. Simplified 56.5%

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

   if -0.309999999999999998 < x < 0.091999999999999998

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

   8. Simplified 98.6%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.092:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right)\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + t\_0\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \sqrt{2}, \sqrt{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(t\_0 + \left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_1
         (/
          (+
           2.0
           (*
            (- (cos x) (cos y))
            (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0)))))
          (* 3.0 (+ (+ 1.0 (* (cos x) (fma 0.5 (sqrt 5.0) -0.5))) t_0)))))
   (if (<= x -0.31)
     t_1
     (if (<= x 0.05)
       (/
        (+
         2.0
         (*
          (fma -0.5 (* x x) (- 1.0 (cos y)))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (fma
            x
            (sqrt 2.0)
            (*
             (sqrt 2.0)
             (fma x (* (* x x) -0.16666666666666666) (* -0.0625 (sin y))))))))
        (*
         3.0
         (+ t_0 (+ 1.0 (* (+ (sqrt 5.0) -1.0) (fma (* x x) -0.25 0.5))))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))))) / (3.0 * ((1.0 + (cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0));
	double tmp;
	if (x <= -0.31) {
		tmp = t_1;
	} else if (x <= 0.05) {
		tmp = (2.0 + (fma(-0.5, (x * x), (1.0 - cos(y))) * ((sin(y) - (sin(x) / 16.0)) * fma(x, sqrt(2.0), (sqrt(2.0) * fma(x, ((x * x) * -0.16666666666666666), (-0.0625 * sin(y)))))))) / (3.0 * (t_0 + (1.0 + ((sqrt(5.0) + -1.0) * fma((x * x), -0.25, 0.5)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * fma(0.5, sqrt(5.0), -0.5))) + t_0)))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_1;
	elseif (x <= 0.05)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * fma(x, sqrt(2.0), Float64(sqrt(2.0) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), Float64(-0.0625 * sin(y)))))))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) * fma(Float64(x * x), -0.25, 0.5))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$1, If[LessEqual[x, 0.05], N[(N[(2.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.050000000000000003 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-sqrt.f64 56.5

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

   8. Simplified 56.5%

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

   if -0.309999999999999998 < x < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.4

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

   8. Simplified 98.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. associate-+l+ N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

   11. Simplified 98.4%

       \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, \sqrt{2}, \sqrt{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, -0.0625 \cdot \sin y\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \sqrt{2}, \sqrt{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_2 := \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t\_1 + \left(1 + \cos x \cdot \frac{t\_0}{2}\right)\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \sqrt{2}, \sqrt{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(t\_1 + \left(1 + t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_2
         (/
          (+
           2.0
           (* (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0))) (+ (cos x) -1.0)))
          (* 3.0 (+ t_1 (+ 1.0 (* (cos x) (/ t_0 2.0))))))))
   (if (<= x -0.31)
     t_2
     (if (<= x 0.05)
       (/
        (+
         2.0
         (*
          (fma -0.5 (* x x) (- 1.0 (cos y)))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (fma
            x
            (sqrt 2.0)
            (*
             (sqrt 2.0)
             (fma x (* (* x x) -0.16666666666666666) (* -0.0625 (sin y))))))))
        (* 3.0 (+ t_1 (+ 1.0 (* t_0 (fma (* x x) -0.25 0.5))))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_2 = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))) * (cos(x) + -1.0))) / (3.0 * (t_1 + (1.0 + (cos(x) * (t_0 / 2.0)))));
	double tmp;
	if (x <= -0.31) {
		tmp = t_2;
	} else if (x <= 0.05) {
		tmp = (2.0 + (fma(-0.5, (x * x), (1.0 - cos(y))) * ((sin(y) - (sin(x) / 16.0)) * fma(x, sqrt(2.0), (sqrt(2.0) * fma(x, ((x * x) * -0.16666666666666666), (-0.0625 * sin(y)))))))) / (3.0 * (t_1 + (1.0 + (t_0 * fma((x * x), -0.25, 0.5)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_2 = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(t_1 + Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))))))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_2;
	elseif (x <= 0.05)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * fma(x, sqrt(2.0), Float64(sqrt(2.0) * fma(x, Float64(Float64(x * x) * -0.16666666666666666), Float64(-0.0625 * sin(y)))))))) / Float64(3.0 * Float64(t_1 + Float64(1.0 + Float64(t_0 * fma(Float64(x * x), -0.25, 0.5))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$2, If[LessEqual[x, 0.05], N[(N[(2.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(1.0 + N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.050000000000000003 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 56.3

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

   8. Simplified 56.3%

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

   if -0.309999999999999998 < x < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.4

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

   8. Simplified 98.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. associate-+l+ N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

   11. Simplified 98.4%

       \[\leadsto \frac{2 + \left(\color{blue}{\mathsf{fma}\left(x, \sqrt{2}, \sqrt{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, -0.0625 \cdot \sin y\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \mathsf{fma}\left(x, \sqrt{2}, \sqrt{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)\\ t_1 := \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(x, 1.00390625, -0.0625 \cdot \sin y\right), x \cdot \left(-0.0625 \cdot x\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))))
        (t_1
         (/
          (+
           2.0
           (* (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0))) (+ (cos x) -1.0)))
          t_0)))
   (if (<= x -0.31)
     t_1
     (if (<= x 0.05)
       (/
        (+
         2.0
         (*
          (fma -0.5 (* x x) (- 1.0 (cos y)))
          (*
           (sqrt 2.0)
           (fma
            (sin y)
            (fma x 1.00390625 (* -0.0625 (sin y)))
            (* x (* -0.0625 x))))))
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))));
	double t_1 = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))) * (cos(x) + -1.0))) / t_0;
	double tmp;
	if (x <= -0.31) {
		tmp = t_1;
	} else if (x <= 0.05) {
		tmp = (2.0 + (fma(-0.5, (x * x), (1.0 - cos(y))) * (sqrt(2.0) * fma(sin(y), fma(x, 1.00390625, (-0.0625 * sin(y))), (x * (-0.0625 * x)))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))))
	t_1 = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))) * Float64(cos(x) + -1.0))) / t_0)
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_1;
	elseif (x <= 0.05)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))) * Float64(sqrt(2.0) * fma(sin(y), fma(x, 1.00390625, Float64(-0.0625 * sin(y))), Float64(x * Float64(-0.0625 * x)))))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$1, If[LessEqual[x, 0.05], N[(N[(2.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(x * 1.00390625 + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(-0.0625 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.050000000000000003 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 56.3

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

   8. Simplified 56.3%

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

   if -0.309999999999999998 < x < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 98.2%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(x, 1.00390625, -0.0625 \cdot \sin y\right), x \cdot \left(x \cdot -0.0625\right)\right)\right)} \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(x, 1.00390625, -0.0625 \cdot \sin y\right), x \cdot \left(-0.0625 \cdot x\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_2 := \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t\_1 + \left(1 + \cos x \cdot \frac{t\_0}{2}\right)\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{3 \cdot \left(t\_1 + \left(1 + t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_2
         (/
          (+
           2.0
           (* (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0))) (+ (cos x) -1.0)))
          (* 3.0 (+ t_1 (+ 1.0 (* (cos x) (/ t_0 2.0))))))))
   (if (<= x -0.31)
     t_2
     (if (<= x 0.05)
       (/
        (+
         2.0
         (*
          (fma -0.5 (* x x) (- 1.0 (cos y)))
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (fma -0.0625 (sin y) x)))))
        (* 3.0 (+ t_1 (+ 1.0 (* t_0 (fma (* x x) -0.25 0.5))))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_2 = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))) * (cos(x) + -1.0))) / (3.0 * (t_1 + (1.0 + (cos(x) * (t_0 / 2.0)))));
	double tmp;
	if (x <= -0.31) {
		tmp = t_2;
	} else if (x <= 0.05) {
		tmp = (2.0 + (fma(-0.5, (x * x), (1.0 - cos(y))) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(-0.0625, sin(y), x))))) / (3.0 * (t_1 + (1.0 + (t_0 * fma((x * x), -0.25, 0.5)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_2 = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(t_1 + Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))))))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_2;
	elseif (x <= 0.05)
		tmp = Float64(Float64(2.0 + Float64(fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))))) / Float64(3.0 * Float64(t_1 + Float64(1.0 + Float64(t_0 * fma(Float64(x * x), -0.25, 0.5))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$2, If[LessEqual[x, 0.05], N[(N[(2.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(1.0 + N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.050000000000000003 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 56.3

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

   8. Simplified 56.3%

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

   if -0.309999999999999998 < x < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.4

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

   8. Simplified 98.4%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sin.f64 98.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Simplified 98.1%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_2 := \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t\_1 + \left(1 + \cos x \cdot \frac{t\_0}{2}\right)\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, x \cdot 1.00390625\right), -0.0625 \cdot \left(x \cdot x\right)\right)\right)}{3 \cdot \left(t\_1 + \left(1 + t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_2
         (/
          (+
           2.0
           (* (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0))) (+ (cos x) -1.0)))
          (* 3.0 (+ t_1 (+ 1.0 (* (cos x) (/ t_0 2.0))))))))
   (if (<= x -0.31)
     t_2
     (if (<= x 0.05)
       (/
        (+
         2.0
         (*
          (- (cos x) (cos y))
          (*
           (sqrt 2.0)
           (fma
            (sin y)
            (fma -0.0625 (sin y) (* x 1.00390625))
            (* -0.0625 (* x x))))))
        (* 3.0 (+ t_1 (+ 1.0 (* t_0 (fma (* x x) -0.25 0.5))))))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_2 = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))) * (cos(x) + -1.0))) / (3.0 * (t_1 + (1.0 + (cos(x) * (t_0 / 2.0)))));
	double tmp;
	if (x <= -0.31) {
		tmp = t_2;
	} else if (x <= 0.05) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * fma(sin(y), fma(-0.0625, sin(y), (x * 1.00390625)), (-0.0625 * (x * x)))))) / (3.0 * (t_1 + (1.0 + (t_0 * fma((x * x), -0.25, 0.5)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_2 = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(t_1 + Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))))))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_2;
	elseif (x <= 0.05)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * fma(sin(y), fma(-0.0625, sin(y), Float64(x * 1.00390625)), Float64(-0.0625 * Float64(x * x)))))) / Float64(3.0 * Float64(t_1 + Float64(1.0 + Float64(t_0 * fma(Float64(x * x), -0.25, 0.5))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$2, If[LessEqual[x, 0.05], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[(x * 1.00390625), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(1.0 + N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.050000000000000003 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 56.3

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

   8. Simplified 56.3%

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

   if -0.309999999999999998 < x < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\frac{-1}{4} \cdot {x}^{2}\right) \cdot \left(\sqrt{5} - 1\right)} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\sqrt{5}} + -1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 98.1

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 98.1%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, x \cdot 1.00390625\right), -0.0625 \cdot \left(x \cdot x\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{t\_1}{2} + \left(1 + \cos x \cdot \frac{t\_0}{2}\right)\right)}\\ \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, 0.5 \cdot t\_1, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), t\_0, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (+
           2.0
           (* (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0))) (+ (cos x) -1.0)))
          (*
           3.0
           (+ (* (cos y) (/ t_1 2.0)) (+ 1.0 (* (cos x) (/ t_0 2.0))))))))
   (if (<= x -0.31)
     t_2
     (if (<= x 0.05)
       (/
        (*
         0.3333333333333333
         (fma
          (sqrt 2.0)
          (*
           (fma -0.5 (* x x) (- 1.0 (cos y)))
           (* (fma -0.0625 (sin y) x) (fma -0.0625 (sin x) (sin y))))
          2.0))
        (fma (cos y) (* 0.5 t_1) (fma (fma (* x x) -0.25 0.5) t_0 1.0)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))) * (cos(x) + -1.0))) / (3.0 * ((cos(y) * (t_1 / 2.0)) + (1.0 + (cos(x) * (t_0 / 2.0)))));
	double tmp;
	if (x <= -0.31) {
		tmp = t_2;
	} else if (x <= 0.05) {
		tmp = (0.3333333333333333 * fma(sqrt(2.0), (fma(-0.5, (x * x), (1.0 - cos(y))) * (fma(-0.0625, sin(y), x) * fma(-0.0625, sin(x), sin(y)))), 2.0)) / fma(cos(y), (0.5 * t_1), fma(fma((x * x), -0.25, 0.5), t_0, 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(t_1 / 2.0)) + Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))))))
	tmp = 0.0
	if (x <= -0.31)
		tmp = t_2;
	elseif (x <= 0.05)
		tmp = Float64(Float64(0.3333333333333333 * fma(sqrt(2.0), Float64(fma(-0.5, Float64(x * x), Float64(1.0 - cos(y))) * Float64(fma(-0.0625, sin(y), x) * fma(-0.0625, sin(x), sin(y)))), 2.0)) / fma(cos(y), Float64(0.5 * t_1), fma(fma(Float64(x * x), -0.25, 0.5), 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] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.31], t$95$2, If[LessEqual[x, 0.05], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.5 * N[(x * x), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + x), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.309999999999999998 or 0.050000000000000003 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.5

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

   5. Simplified 56.5%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 56.3

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

   8. Simplified 56.3%

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

   if -0.309999999999999998 < x < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.4

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

   8. Simplified 98.4%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-sin.f64 98.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\sin y}, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Simplified 98.1%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   12. Taylor expanded in y around inf

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

   14. Simplified 98.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} + -1, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.31:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right) \cdot \left(\mathsf{fma}\left(-0.0625, \sin y, x\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right)\right), 2\right)}{\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 0.5\right), \sqrt{5} + -1, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_1 := \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t\_0 + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{if}\;x \leq -6.1 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot \mathsf{fma}\left(x, 1.00390625, -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(t\_0 + \left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_1
         (/
          (+
           2.0
           (* (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0))) (+ (cos x) -1.0)))
          (* 3.0 (+ t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))))))
   (if (<= x -6.1e-5)
     t_1
     (if (<= x 0.05)
       (/
        (+
         2.0
         (*
          (- (cos x) (cos y))
          (* (sqrt 2.0) (* (sin y) (fma x 1.00390625 (* -0.0625 (sin y)))))))
        (* 3.0 (+ t_0 (+ 1.0 (fma 0.5 (sqrt 5.0) -0.5)))))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_1 = (2.0 + ((sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))) * (cos(x) + -1.0))) / (3.0 * (t_0 + (1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0)))));
	double tmp;
	if (x <= -6.1e-5) {
		tmp = t_1;
	} else if (x <= 0.05) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (sin(y) * fma(x, 1.00390625, (-0.0625 * sin(y))))))) / (3.0 * (t_0 + (1.0 + fma(0.5, sqrt(5.0), -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_1 = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))))))
	tmp = 0.0
	if (x <= -6.1e-5)
		tmp = t_1;
	elseif (x <= 0.05)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(sin(y) * fma(x, 1.00390625, Float64(-0.0625 * sin(y))))))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + fma(0.5, sqrt(5.0), -0.5)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.1e-5], t$95$1, If[LessEqual[x, 0.05], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(x * 1.00390625 + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.09999999999999987e-5 or 0.050000000000000003 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 56.3

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

   5. Simplified 56.3%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 56.2

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\color{blue}{\cos x} + -1\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 56.2%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \color{blue}{\left(\cos x + -1\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -6.09999999999999987e-5 < x < 0.050000000000000003

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right) + \frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}} + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + x \cdot \color{blue}{\left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \sqrt{2}\right)}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \sqrt{2}}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)} + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot x\right) \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\sqrt{2} \cdot \left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 99.1%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 98.8

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 98.8%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + \frac{257}{256} \cdot \left(x \cdot \left(\sin y \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}} + \frac{257}{256} \cdot \left(x \cdot \left(\sin y \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       2. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(\frac{257}{256} \cdot x\right) \cdot \left(\sin y \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(\left(\frac{257}{256} \cdot x\right) \cdot \sin y\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\frac{-1}{16} \cdot \color{blue}{\left(\sin y \cdot \sin y\right)}\right) \cdot \sqrt{2} + \left(\left(\frac{257}{256} \cdot x\right) \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       5. associate-*r* N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\left(\frac{-1}{16} \cdot \sin y\right) \cdot \sin y\right)} \cdot \sqrt{2} + \left(\left(\frac{257}{256} \cdot x\right) \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{2 + \left(\color{blue}{\left(\sin y \cdot \left(\frac{-1}{16} \cdot \sin y\right)\right)} \cdot \sqrt{2} + \left(\left(\frac{257}{256} \cdot x\right) \cdot \sin y\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{2 + \left(\left(\sin y \cdot \left(\frac{-1}{16} \cdot \sin y\right)\right) \cdot \sqrt{2} + \color{blue}{\left(\sin y \cdot \left(\frac{257}{256} \cdot x\right)\right)} \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       8. distribute-rgt-in N/A

          \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin y \cdot \left(\frac{-1}{16} \cdot \sin y\right) + \sin y \cdot \left(\frac{257}{256} \cdot x\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       9. distribute-lft-in N/A

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\left(\sin y \cdot \left(\frac{-1}{16} \cdot \sin y + \frac{257}{256} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin y \cdot \left(\frac{-1}{16} \cdot \sin y + \frac{257}{256} \cdot x\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2}} \cdot \left(\sin y \cdot \left(\frac{-1}{16} \cdot \sin y + \frac{257}{256} \cdot x\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \color{blue}{\left(\sin y \cdot \left(\frac{-1}{16} \cdot \sin y + \frac{257}{256} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       13. lower-sin.f64 N/A

           \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\color{blue}{\sin y} \cdot \left(\frac{-1}{16} \cdot \sin y + \frac{257}{256} \cdot x\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       14. +-commutative N/A

           \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot \color{blue}{\left(\frac{257}{256} \cdot x + \frac{-1}{16} \cdot \sin y\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       15. *-commutative N/A

           \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot \left(\color{blue}{x \cdot \frac{257}{256}} + \frac{-1}{16} \cdot \sin y\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       16. lower-fma.f64 N/A

           \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot \color{blue}{\mathsf{fma}\left(x, \frac{257}{256}, \frac{-1}{16} \cdot \sin y\right)}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin y \cdot \mathsf{fma}\left(x, \frac{257}{256}, \color{blue}{\frac{-1}{16} \cdot \sin y}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. Simplified 98.9%

       \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\sin y \cdot \mathsf{fma}\left(x, 1.00390625, -0.0625 \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\sin y \cdot \mathsf{fma}\left(x, 1.00390625, -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+ 2.0 (* (- (cos x) (cos y)) (* (* (sqrt 2.0) -0.0625) (pow (sin y) 2.0))))
  (*
   3.0
   (+
    (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))
    (+ 1.0 (fma 0.5 (sqrt 5.0) -0.5))))))

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * -0.0625) * pow(sin(y), 2.0)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + fma(0.5, sqrt(5.0), -0.5))));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * -0.0625) * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + fma(0.5, sqrt(5.0), -0.5)))))
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right) + \frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}} + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + x \cdot \color{blue}{\left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \sqrt{2}\right)}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \sqrt{2}}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. distribute-rgt-out N/A

      \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)} + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot x\right) \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\sqrt{2} \cdot \left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 56.7%

   \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-sqrt.f64 56.3

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Simplified 56.3%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \color{blue}{\left(\sqrt{2} \cdot {\sin y}^{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

    2. associate-*r* N/A

       \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot {\sin y}^{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \sqrt{2}\right) \cdot {\sin y}^{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

    4. *-commutative N/A

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right)} \cdot {\sin y}^{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \frac{-1}{16}\right)} \cdot {\sin y}^{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

    6. lower-sqrt.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\color{blue}{\sqrt{2}} \cdot \frac{-1}{16}\right) \cdot {\sin y}^{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

    7. lower-pow.f64 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \frac{-1}{16}\right) \cdot \color{blue}{{\sin y}^{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

    8. lower-sin.f64 63.5

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\color{blue}{\sin y}}^{2}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

11. Simplified 63.5%

    \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

12. Final simplification 63.5%

    \[\leadsto \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)\right)\right)} \]
13. Add Preprocessing

Alternative 18: 59.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   0.3333333333333333
   (* (* (sqrt 2.0) (pow (sin y) 2.0)) (* -0.0625 (- 1.0 (cos y))))
   0.6666666666666666)
  (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (+ (sqrt 5.0) -1.0)) 1.0)))

C

double code(double x, double y) {
	return fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(y), 2.0)) * (-0.0625 * (1.0 - cos(y)))), 0.6666666666666666) / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), (sqrt(5.0) + -1.0)), 1.0);
}

Julia

function code(x, y)
	return Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(y) ^ 2.0)) * Float64(-0.0625 * Float64(1.0 - cos(y)))), 0.6666666666666666) / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(sqrt(5.0) + -1.0)), 1.0))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. associate--l+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + \left(1 - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, \color{blue}{1 - \cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-cos.f64 57.0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 57.0%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\frac{-1}{4} \cdot {x}^{2}\right) \cdot \left(\sqrt{5} - 1\right)} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. sub-neg N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\sqrt{5}} + -1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-*.f64 56.3

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Simplified 56.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

11. Simplified 63.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot -0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]

12. Final simplification 63.4%

    \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \]
13. Add Preprocessing

Alternative 19: 59.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   0.3333333333333333
   (* (* (sqrt 2.0) (pow (sin y) 2.0)) (* -0.0625 (- 1.0 (cos y))))
   0.6666666666666666)
  (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) 0.5)))

C

double code(double x, double y) {
	return fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(y), 2.0)) * (-0.0625 * (1.0 - cos(y)))), 0.6666666666666666) / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), 0.5);
}

Julia

function code(x, y)
	return Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(y) ^ 2.0)) * Float64(-0.0625 * Float64(1.0 - cos(y)))), 0.6666666666666666) / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), 0.5))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right) + \frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}} + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + x \cdot \color{blue}{\left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \sqrt{2}\right)}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \sqrt{2}}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. distribute-rgt-out N/A

      \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)} + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot x\right) \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\sqrt{2} \cdot \left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 56.7%

   \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-sqrt.f64 56.3

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Simplified 56.3%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
10. 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

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]

11. Simplified 63.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot -0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)}} \]

12. Final simplification 63.4%

    \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), 0.5\right)} \]
13. Add Preprocessing

Alternative 20: 40.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, 2, 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   0.3333333333333333
   (* (* (sqrt 2.0) (pow (sin y) 2.0)) (* -0.0625 (- 1.0 (cos y))))
   0.6666666666666666)
  (fma 0.5 2.0 1.0)))

C

double code(double x, double y) {
	return fma(0.3333333333333333, ((sqrt(2.0) * pow(sin(y), 2.0)) * (-0.0625 * (1.0 - cos(y)))), 0.6666666666666666) / fma(0.5, 2.0, 1.0);
}

Julia

function code(x, y)
	return Float64(fma(0.3333333333333333, Float64(Float64(sqrt(2.0) * (sin(y) ^ 2.0)) * Float64(-0.0625 * Float64(1.0 - cos(y)))), 0.6666666666666666) / fma(0.5, 2.0, 1.0))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. associate--l+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + \left(1 - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, \color{blue}{1 - \cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-cos.f64 57.0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 57.0%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\frac{-1}{4} \cdot {x}^{2}\right) \cdot \left(\sqrt{5} - 1\right)} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. sub-neg N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\sqrt{5}} + -1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   11. lower-*.f64 56.3

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Simplified 56.3%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

11. Simplified 63.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot -0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]

12. Taylor expanded in y around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{2}, 1\right)} \]
13. Step-by-step derivation

14. Simplified 44.0%

    \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot -0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \color{blue}{2}, 1\right)} \]

15. Final simplification 44.0%

    \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, 2, 1\right)} \]
16. Add Preprocessing

Alternative 21: 36.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \frac{\mathsf{fma}\left(0.3333333333333333, \left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + t\_0, 0.5\right)}\\ \mathbf{if}\;y \leq -7.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(-0.0625 \cdot \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \left(y \cdot y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \sqrt{5} + -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (/
          (fma
           0.3333333333333333
           (* (* x x) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
           0.6666666666666666)
          (fma 0.5 (+ (sqrt 5.0) t_0) 0.5))))
   (if (<= y -7.1)
     t_1
     (if (<= y 3.2)
       (/
        (fma
         0.3333333333333333
         (* (* -0.0625 (- 1.0 (cos y))) (* (sqrt 2.0) (* y y)))
         0.6666666666666666)
        (fma 0.5 (fma (cos y) t_0 (+ (sqrt 5.0) -1.0)) 1.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = fma(0.3333333333333333, ((x * x) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, (sqrt(5.0) + t_0), 0.5);
	double tmp;
	if (y <= -7.1) {
		tmp = t_1;
	} else if (y <= 3.2) {
		tmp = fma(0.3333333333333333, ((-0.0625 * (1.0 - cos(y))) * (sqrt(2.0) * (y * y))), 0.6666666666666666) / fma(0.5, fma(cos(y), t_0, (sqrt(5.0) + -1.0)), 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(0.3333333333333333, Float64(Float64(x * x) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, Float64(sqrt(5.0) + t_0), 0.5))
	tmp = 0.0
	if (y <= -7.1)
		tmp = t_1;
	elseif (y <= 3.2)
		tmp = Float64(fma(0.3333333333333333, Float64(Float64(-0.0625 * Float64(1.0 - cos(y))) * Float64(sqrt(2.0) * Float64(y * y))), 0.6666666666666666) / fma(0.5, fma(cos(y), t_0, Float64(sqrt(5.0) + -1.0)), 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[(0.3333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + t$95$0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.1], t$95$1, If[LessEqual[y, 3.2], N[(N[(0.3333333333333333 * N[(N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0999999999999996 or 3.2000000000000002 < y

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right) + \frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}} + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + x \cdot \color{blue}{\left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \sqrt{2}\right)}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \sqrt{2}}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)} + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot x\right) \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\sqrt{2} \cdot \left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 52.0%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-sqrt.f64 51.6

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 51.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   11. Simplified 14.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)}} \]

   if -7.0999999999999996 < y < 3.2000000000000002

   1. Initial program 99.6%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {x}^{2}\right) - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. associate--l+ N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + \left(1 - \cos y\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, \color{blue}{1 - \cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-cos.f64 61.6

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \color{blue}{\cos y}\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Simplified 61.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{-1}{4} \cdot \left({x}^{2} \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\left(\frac{-1}{4} \cdot {x}^{2}\right) \cdot \left(\sqrt{5} - 1\right)} + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. distribute-rgt-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} - 1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + \color{blue}{-1}\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right)} \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\color{blue}{\sqrt{5}} + -1\right) \cdot \left(\frac{-1}{4} \cdot {x}^{2} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{4}, \frac{1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{4}, \frac{1}{2}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      11. lower-*.f64 60.1

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.25, 0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. Simplified 60.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(-0.5, x \cdot x, 1 - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{5} + -1\right) \cdot \mathsf{fma}\left(x \cdot x, -0.25, 0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)}} \]

   11. Simplified 67.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(\sqrt{2} \cdot {\sin y}^{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot -0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}} \]

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\left({y}^{2} \cdot \sqrt{2}\right)} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\left({y}^{2} \cdot \sqrt{2}\right)} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \left(\color{blue}{\left(y \cdot y\right)} \cdot \sqrt{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \left(\color{blue}{\left(y \cdot y\right)} \cdot \sqrt{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \]

       4. lower-sqrt.f64 68.0

          \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \left(\left(y \cdot y\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot \left(\left(1 - \cos y\right) \cdot -0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \]

   14. Simplified 68.0%

       \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{\left(\left(y \cdot y\right) \cdot \sqrt{2}\right)} \cdot \left(\left(1 - \cos y\right) \cdot -0.0625\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)}\\ \mathbf{elif}\;y \leq 3.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(-0.0625 \cdot \left(1 - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \left(y \cdot y\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 32.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.3333333333333333, \left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   0.3333333333333333
   (* (* x x) (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625)))
   0.6666666666666666)
  (fma 0.5 (+ (sqrt 5.0) (- 3.0 (sqrt 5.0))) 0.5)))

C

double code(double x, double y) {
	return fma(0.3333333333333333, ((x * x) * (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, (sqrt(5.0) + (3.0 - sqrt(5.0))), 0.5);
}

Julia

function code(x, y)
	return Float64(fma(0.3333333333333333, Float64(Float64(x * x) * Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, Float64(sqrt(5.0) + Float64(3.0 - sqrt(5.0))), 0.5))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right) + \frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}} + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + x \cdot \color{blue}{\left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \sqrt{2}\right)}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \sqrt{2}}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. distribute-rgt-out N/A

      \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)} + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot x\right) \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\sqrt{2} \cdot \left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 56.7%

   \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-sqrt.f64 56.3

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Simplified 56.3%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

11. Simplified 37.1%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)}} \]
12. Add Preprocessing

Alternative 23: 32.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left(0.03125 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   0.3333333333333333
   (* (sqrt 2.0) (* 0.03125 (* (* x x) (* x x))))
   0.6666666666666666)
  (fma 0.5 (+ (sqrt 5.0) (- 3.0 (sqrt 5.0))) 0.5)))

C

double code(double x, double y) {
	return fma(0.3333333333333333, (sqrt(2.0) * (0.03125 * ((x * x) * (x * x)))), 0.6666666666666666) / fma(0.5, (sqrt(5.0) + (3.0 - sqrt(5.0))), 0.5);
}

Julia

function code(x, y)
	return Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64(0.03125 * Float64(Float64(x * x) * Float64(x * x)))), 0.6666666666666666) / fma(0.5, Float64(sqrt(5.0) + Float64(3.0 - sqrt(5.0))), 0.5))
end

Wolfram

code[x_, y_] := N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.03125 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \color{blue}{\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right) + \frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-+r+ N/A

      \[\leadsto \frac{2 + \color{blue}{\left(\left(\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2}} + x \cdot \left(\sqrt{2} \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + x \cdot \color{blue}{\left(\left(\sin y + \frac{1}{256} \cdot \sin y\right) \cdot \sqrt{2}\right)}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\left(\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) \cdot \sqrt{2}}\right) + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. distribute-rgt-out N/A

      \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right)} + x \cdot \left(\frac{-1}{16} \cdot \left(x \cdot \sqrt{2}\right)\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   8. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + x \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot x\right) \cdot \sqrt{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\frac{-1}{16} \cdot {\sin y}^{2} + x \cdot \left(\sin y + \frac{1}{256} \cdot \sin y\right)\right) + \color{blue}{\sqrt{2} \cdot \left(x \cdot \left(\frac{-1}{16} \cdot x\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Simplified 56.7%

   \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} - 1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot -1\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(\frac{-1}{16}, \sin y, \frac{257}{256} \cdot x\right), \left(x \cdot x\right) \cdot \frac{-1}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. lower-sqrt.f64 56.3

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \mathsf{fma}\left(0.5, \color{blue}{\sqrt{5}}, -0.5\right)\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Simplified 56.3%

   \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \mathsf{fma}\left(\sin y, \mathsf{fma}\left(-0.0625, \sin y, 1.00390625 \cdot x\right), \left(x \cdot x\right) \cdot -0.0625\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right)}\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left(2 + \frac{-1}{16} \cdot \left({x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\frac{1}{2} + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

11. Simplified 37.1%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)}} \]

12. Taylor expanded in x around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\frac{1}{32} \cdot \left({x}^{4} \cdot \sqrt{2}\right)}, \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{32} \cdot {x}^{4}\right) \cdot \sqrt{2}}, \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{32} \cdot {x}^{4}\right)}, \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{32} \cdot {x}^{4}\right)}, \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    4. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{32} \cdot {x}^{4}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt{2} \cdot \color{blue}{\left(\frac{1}{32} \cdot {x}^{4}\right)}, \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    6. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt{2} \cdot \left(\frac{1}{32} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    7. pow-sqr N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt{2} \cdot \left(\frac{1}{32} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt{2} \cdot \left(\frac{1}{32} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    9. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt{2} \cdot \left(\frac{1}{32} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right)\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt{2} \cdot \left(\frac{1}{32} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right)\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    11. unpow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt{2} \cdot \left(\frac{1}{32} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right), \frac{2}{3}\right)}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5} + \left(3 - \sqrt{5}\right), \frac{1}{2}\right)} \]

    12. lower-*.f64 36.8

        \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left(0.03125 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)} \]

14. Simplified 36.8%

    \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{\sqrt{2} \cdot \left(0.03125 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}, 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \sqrt{5} + \left(3 - \sqrt{5}\right), 0.5\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))))