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

Percentage Accurate: 99.3% → 99.4%

Time: 23.2s

Alternatives: 27

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)} \end{array} \]

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))))
  (fma
   1.5
   (fma (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))) (* (cos x) (+ (sqrt 5.0) -1.0)))
   3.0)))

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)))) / fma(1.5, fma(cos(y), (4.0 / (3.0 + sqrt(5.0))), (cos(x) * (sqrt(5.0) + -1.0))), 3.0);
}

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)))) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(3.0 + sqrt(5.0))), Float64(cos(x) * Float64(sqrt(5.0) + -1.0))), 3.0))
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[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

8. Final simplification 99.5%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-sin.f64 57.5

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

10. Applied rewrites 57.5%

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

11. Taylor expanded in y around inf

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

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

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

    4. metadata-eval N/A

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

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

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

    6. associate-*r* N/A

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

    7. associate-*r* N/A

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

    8. lower-fma.f64 N/A

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

13. Applied rewrites 99.4%

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

Alternative 6: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(\sqrt{5} + -1\right)\\ t_1 := \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -0.44:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ (sqrt 5.0) -1.0)))
        (t_1
         (/
          (fma
           (- (cos x) (cos y))
           (* (fma -0.0625 (sin x) (sin y)) (* (sqrt 2.0) (sin x)))
           2.0)
          (fma 1.5 (fma (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))) t_0) 3.0))))
   (if (<= x -0.44)
     t_1
     (if (<= x 7.5e-12)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ (sin x) 16.0)))
          (fma
           (* x x)
           (fma
            (* x x)
            (fma (* x x) -0.001388888888888889 0.041666666666666664)
            -0.5)
           (- 1.0 (cos y)))))
        (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_0) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sqrt(5.0) + -1.0);
	double t_1 = fma((cos(x) - cos(y)), (fma(-0.0625, sin(x), sin(y)) * (sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (3.0 + sqrt(5.0))), t_0), 3.0);
	double tmp;
	if (x <= -0.44) {
		tmp = t_1;
	} else if (x <= 7.5e-12) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), (1.0 - cos(y))))) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), t_0), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sqrt(5.0) + -1.0))
	t_1 = Float64(fma(Float64(cos(x) - cos(y)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(3.0 + sqrt(5.0))), t_0), 3.0))
	tmp = 0.0
	if (x <= -0.44)
		tmp = t_1;
	elseif (x <= 7.5e-12)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), Float64(1.0 - cos(y))))) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_0), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.44], t$95$1, If[LessEqual[x, 7.5e-12], 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[(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] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.440000000000000002 or 7.5e-12 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 64.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 64.7

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 64.7%

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

   if -0.440000000000000002 < x < 7.5e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-cos.f64 99.5

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \color{blue}{\cos y}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Applied rewrites 99.5%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.44:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -0.44:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1
         (/
          (fma
           (- (cos x) (cos y))
           (* (fma -0.0625 (sin x) (sin y)) (* (sqrt 2.0) (sin x)))
           2.0)
          (fma
           1.5
           (fma (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))) (* (cos x) t_0))
           3.0))))
   (if (<= x -0.44)
     t_1
     (if (<= x 7.5e-12)
       (/
        (fma
         (sqrt 2.0)
         (*
          (fma (sin y) -0.0625 (sin x))
          (*
           (fma (sin x) -0.0625 (sin y))
           (fma
            (* x x)
            (fma
             x
             (* x (fma (* x x) -0.001388888888888889 0.041666666666666664))
             -0.5)
            (- 1.0 (cos y)))))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* (cos y) (- 3.0 (sqrt 5.0)))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = fma((cos(x) - cos(y)), (fma(-0.0625, sin(x), sin(y)) * (sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (3.0 + sqrt(5.0))), (cos(x) * t_0)), 3.0);
	double tmp;
	if (x <= -0.44) {
		tmp = t_1;
	} else if (x <= 7.5e-12) {
		tmp = fma(sqrt(2.0), (fma(sin(y), -0.0625, sin(x)) * (fma(sin(x), -0.0625, sin(y)) * fma((x * x), fma(x, (x * fma((x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), (1.0 - cos(y))))), 2.0) / fma(1.5, fma(cos(x), t_0, (cos(y) * (3.0 - sqrt(5.0)))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(fma(Float64(cos(x) - cos(y)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(3.0 + sqrt(5.0))), Float64(cos(x) * t_0)), 3.0))
	tmp = 0.0
	if (x <= -0.44)
		tmp = t_1;
	elseif (x <= 7.5e-12)
		tmp = Float64(fma(sqrt(2.0), Float64(fma(sin(y), -0.0625, sin(x)) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), Float64(1.0 - cos(y))))), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(cos(y) * Float64(3.0 - sqrt(5.0)))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.44], t$95$1, If[LessEqual[x, 7.5e-12], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.440000000000000002 or 7.5e-12 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 64.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 64.7

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 64.7%

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

   if -0.440000000000000002 < x < 7.5e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-cos.f64 99.5

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \color{blue}{\cos y}\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)} \]

   10. Applied rewrites 99.5%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.44:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), x, 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1
         (/
          (fma
           (- (cos x) (cos y))
           (* (fma -0.0625 (sin x) (sin y)) (* (sqrt 2.0) (sin x)))
           2.0)
          (fma
           1.5
           (fma (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))) (* (cos x) t_0))
           3.0))))
   (if (<= x -0.2)
     t_1
     (if (<= x 7.5e-12)
       (/
        (fma
         (sqrt 2.0)
         (*
          (fma (sin y) -0.0625 (sin x))
          (*
           (fma (sin x) -0.0625 (sin y))
           (fma
            (* x (fma (* x x) 0.041666666666666664 -0.5))
            x
            (- 1.0 (cos y)))))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* (cos y) (- 3.0 (sqrt 5.0)))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = fma((cos(x) - cos(y)), (fma(-0.0625, sin(x), sin(y)) * (sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (3.0 + sqrt(5.0))), (cos(x) * t_0)), 3.0);
	double tmp;
	if (x <= -0.2) {
		tmp = t_1;
	} else if (x <= 7.5e-12) {
		tmp = fma(sqrt(2.0), (fma(sin(y), -0.0625, sin(x)) * (fma(sin(x), -0.0625, sin(y)) * fma((x * fma((x * x), 0.041666666666666664, -0.5)), x, (1.0 - cos(y))))), 2.0) / fma(1.5, fma(cos(x), t_0, (cos(y) * (3.0 - sqrt(5.0)))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(fma(Float64(cos(x) - cos(y)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(3.0 + sqrt(5.0))), Float64(cos(x) * t_0)), 3.0))
	tmp = 0.0
	if (x <= -0.2)
		tmp = t_1;
	elseif (x <= 7.5e-12)
		tmp = Float64(fma(sqrt(2.0), Float64(fma(sin(y), -0.0625, sin(x)) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(Float64(x * fma(Float64(x * x), 0.041666666666666664, -0.5)), x, Float64(1.0 - cos(y))))), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(cos(y) * Float64(3.0 - sqrt(5.0)))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.2], t$95$1, If[LessEqual[x, 7.5e-12], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.20000000000000001 or 7.5e-12 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 64.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 64.7

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 64.7%

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

   if -0.20000000000000001 < x < 7.5e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-cos.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right) \cdot x, x, 1 - \color{blue}{\cos y}\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)} \]

   10. Applied rewrites 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right) \cdot x, x, 1 - \cos y\right)}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), x, 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1
         (/
          (fma
           (- (cos x) (cos y))
           (* (fma -0.0625 (sin x) (sin y)) (* (sqrt 2.0) (sin x)))
           2.0)
          (fma
           1.5
           (fma (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))) (* (cos x) t_0))
           3.0))))
   (if (<= x -0.2)
     t_1
     (if (<= x 7.5e-12)
       (/
        (fma
         (sqrt 2.0)
         (*
          (fma (sin y) -0.0625 (sin x))
          (* (fma (sin x) -0.0625 (sin y)) (fma x (* x -0.5) (- 1.0 (cos y)))))
         2.0)
        (fma 1.5 (fma (cos x) t_0 (* (cos y) (- 3.0 (sqrt 5.0)))) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = fma((cos(x) - cos(y)), (fma(-0.0625, sin(x), sin(y)) * (sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (3.0 + sqrt(5.0))), (cos(x) * t_0)), 3.0);
	double tmp;
	if (x <= -0.2) {
		tmp = t_1;
	} else if (x <= 7.5e-12) {
		tmp = fma(sqrt(2.0), (fma(sin(y), -0.0625, sin(x)) * (fma(sin(x), -0.0625, sin(y)) * fma(x, (x * -0.5), (1.0 - cos(y))))), 2.0) / fma(1.5, fma(cos(x), t_0, (cos(y) * (3.0 - sqrt(5.0)))), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(fma(Float64(cos(x) - cos(y)), Float64(fma(-0.0625, sin(x), sin(y)) * Float64(sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(3.0 + sqrt(5.0))), Float64(cos(x) * t_0)), 3.0))
	tmp = 0.0
	if (x <= -0.2)
		tmp = t_1;
	elseif (x <= 7.5e-12)
		tmp = Float64(fma(sqrt(2.0), Float64(fma(sin(y), -0.0625, sin(x)) * Float64(fma(sin(x), -0.0625, sin(y)) * fma(x, Float64(x * -0.5), Float64(1.0 - cos(y))))), 2.0) / fma(1.5, fma(cos(x), t_0, Float64(cos(y) * Float64(3.0 - sqrt(5.0)))), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.2], t$95$1, If[LessEqual[x, 7.5e-12], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.5), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.20000000000000001 or 7.5e-12 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 64.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 64.7

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 64.7%

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

   if -0.20000000000000001 < x < 7.5e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 99.3

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

   10. Applied rewrites 99.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \mathsf{fma}\left(x, x \cdot -0.5, 1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + t\_0 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right), x, \sqrt{5}\right) + \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (/
          (fma
           t_0
           (* (fma -0.0625 (sin x) (sin y)) (* (sqrt 2.0) (sin x)))
           2.0)
          (fma
           1.5
           (fma
            (cos y)
            (/ 4.0 (+ 3.0 (sqrt 5.0)))
            (* (cos x) (+ (sqrt 5.0) -1.0)))
           3.0))))
   (if (<= x -0.2)
     t_1
     (if (<= x 7.5e-12)
       (/
        (+
         2.0
         (*
          t_0
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (fma -0.0625 (sin y) x)))))
        (fma
         1.5
         (+
          (fma (* x (fma -0.5 (sqrt 5.0) 0.5)) x (sqrt 5.0))
          (fma (cos y) (- 3.0 (sqrt 5.0)) -1.0))
         3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma(t_0, (fma(-0.0625, sin(x), sin(y)) * (sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (3.0 + sqrt(5.0))), (cos(x) * (sqrt(5.0) + -1.0))), 3.0);
	double tmp;
	if (x <= -0.2) {
		tmp = t_1;
	} else if (x <= 7.5e-12) {
		tmp = (2.0 + (t_0 * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(-0.0625, sin(y), x))))) / fma(1.5, (fma((x * fma(-0.5, sqrt(5.0), 0.5)), x, sqrt(5.0)) + fma(cos(y), (3.0 - sqrt(5.0)), -1.0)), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(fma(t_0, Float64(fma(-0.0625, sin(x), sin(y)) * Float64(sqrt(2.0) * sin(x))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(3.0 + sqrt(5.0))), Float64(cos(x) * Float64(sqrt(5.0) + -1.0))), 3.0))
	tmp = 0.0
	if (x <= -0.2)
		tmp = t_1;
	elseif (x <= 7.5e-12)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))))) / fma(1.5, Float64(fma(Float64(x * fma(-0.5, sqrt(5.0), 0.5)), x, sqrt(5.0)) + fma(cos(y), Float64(3.0 - sqrt(5.0)), -1.0)), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.2], t$95$1, If[LessEqual[x, 7.5e-12], N[(N[(2.0 + N[(t$95$0 * 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[(1.5 * N[(N[(N[(x * N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.20000000000000001 or 7.5e-12 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 64.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 64.7

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\cos x - \cos y, \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 64.7%

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

   if -0.20000000000000001 < x < 7.5e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.2%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right) \cdot x, x, \sqrt{5}\right) + \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, -1\right)}, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right), x, \sqrt{5}\right) + \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x - \cos y, \mathsf{fma}\left(-0.0625, \sin x, \sin y\right) \cdot \left(\sqrt{2} \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \sqrt{2} \cdot \sin x, 2\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot t\_2\right), 3\right)\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;t\_1 \cdot \frac{1}{t\_3}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + t\_0 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right), x, \sqrt{5}\right) + \mathsf{fma}\left(\cos y, t\_2, -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (fma
          (* t_0 (fma -0.0625 (sin x) (sin y)))
          (* (sqrt 2.0) (sin x))
          2.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma 1.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) t_2)) 3.0)))
   (if (<= x -0.2)
     (* t_1 (/ 1.0 t_3))
     (if (<= x 7.5e-12)
       (/
        (+
         2.0
         (*
          t_0
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (fma -0.0625 (sin y) x)))))
        (fma
         1.5
         (+
          (fma (* x (fma -0.5 (sqrt 5.0) 0.5)) x (sqrt 5.0))
          (fma (cos y) t_2 -1.0))
         3.0))
       (/ t_1 t_3)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = fma((t_0 * fma(-0.0625, sin(x), sin(y))), (sqrt(2.0) * sin(x)), 2.0);
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * t_2)), 3.0);
	double tmp;
	if (x <= -0.2) {
		tmp = t_1 * (1.0 / t_3);
	} else if (x <= 7.5e-12) {
		tmp = (2.0 + (t_0 * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(-0.0625, sin(y), x))))) / fma(1.5, (fma((x * fma(-0.5, sqrt(5.0), 0.5)), x, sqrt(5.0)) + fma(cos(y), t_2, -1.0)), 3.0);
	} else {
		tmp = t_1 / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = fma(Float64(t_0 * fma(-0.0625, sin(x), sin(y))), Float64(sqrt(2.0) * sin(x)), 2.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * t_2)), 3.0)
	tmp = 0.0
	if (x <= -0.2)
		tmp = Float64(t_1 * Float64(1.0 / t_3));
	elseif (x <= 7.5e-12)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))))) / fma(1.5, Float64(fma(Float64(x * fma(-0.5, sqrt(5.0), 0.5)), x, sqrt(5.0)) + fma(cos(y), t_2, -1.0)), 3.0));
	else
		tmp = Float64(t_1 / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[x, -0.2], N[(t$95$1 * N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-12], N[(N[(2.0 + N[(t$95$0 * 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[(1.5 * N[(N[(N[(x * N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.20000000000000001

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 67.9

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 67.9%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 67.9%

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

   if -0.20000000000000001 < x < 7.5e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.2%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right) \cdot x, x, \sqrt{5}\right) + \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, -1\right)}, 3\right)} \]

   if 7.5e-12 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 61.9

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 61.9%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 61.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \sqrt{2} \cdot \sin x, 2\right) \cdot \frac{1}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right), x, \sqrt{5}\right) + \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \sqrt{2} \cdot \sin x, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \sqrt{2} \cdot \sin x, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + t\_0 \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right), x, \sqrt{5}\right) + \mathsf{fma}\left(\cos y, t\_1, -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* t_0 (fma -0.0625 (sin x) (sin y)))
           (* (sqrt 2.0) (sin x))
           2.0)
          (fma 1.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (* (cos y) t_1)) 3.0))))
   (if (<= x -0.2)
     t_2
     (if (<= x 7.5e-12)
       (/
        (+
         2.0
         (*
          t_0
          (*
           (- (sin y) (/ (sin x) 16.0))
           (* (sqrt 2.0) (fma -0.0625 (sin y) x)))))
        (fma
         1.5
         (+
          (fma (* x (fma -0.5 (sqrt 5.0) 0.5)) x (sqrt 5.0))
          (fma (cos y) t_1 -1.0))
         3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((t_0 * fma(-0.0625, sin(x), sin(y))), (sqrt(2.0) * sin(x)), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), (cos(y) * t_1)), 3.0);
	double tmp;
	if (x <= -0.2) {
		tmp = t_2;
	} else if (x <= 7.5e-12) {
		tmp = (2.0 + (t_0 * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * fma(-0.0625, sin(y), x))))) / fma(1.5, (fma((x * fma(-0.5, sqrt(5.0), 0.5)), x, sqrt(5.0)) + fma(cos(y), t_1, -1.0)), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64(t_0 * fma(-0.0625, sin(x), sin(y))), Float64(sqrt(2.0) * sin(x)), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(cos(y) * t_1)), 3.0))
	tmp = 0.0
	if (x <= -0.2)
		tmp = t_2;
	elseif (x <= 7.5e-12)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * fma(-0.0625, sin(y), x))))) / fma(1.5, Float64(fma(Float64(x * fma(-0.5, sqrt(5.0), 0.5)), x, sqrt(5.0)) + fma(cos(y), t_1, -1.0)), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.2], t$95$2, If[LessEqual[x, 7.5e-12], N[(N[(2.0 + N[(t$95$0 * 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[(1.5 * N[(N[(N[(x * N[(-0.5 * N[Sqrt[5.0], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.20000000000000001 or 7.5e-12 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 64.7

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\sin x}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 64.7%

       \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \sin x\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 64.7%

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

   if -0.20000000000000001 < x < 7.5e-12

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

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

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 99.2

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.2%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right) \cdot x, x, \sqrt{5}\right) + \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, -1\right)}, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \sqrt{2} \cdot \sin x, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(-0.0625, \sin y, x\right)\right)\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(-0.5, \sqrt{5}, 0.5\right), x, \sqrt{5}\right) + \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\cos x - \cos y\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), \sqrt{2} \cdot \sin x, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -0.0059:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_0 \cdot \left(\left(\cos x + -1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_0 \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_1\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= y -0.0059)
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (* (sqrt 2.0) -0.0625) (pow (sin y) 2.0))))
      (fma 1.5 (fma (cos y) t_1 (* (cos x) t_2)) 3.0))
     (if (<= y 5.4e-5)
       (/
        (fma
         (sqrt 2.0)
         (* t_0 (* (+ (cos x) -1.0) (fma -0.0625 (sin x) y)))
         2.0)
        (fma 1.5 (fma (cos x) t_2 (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))))) 3.0))
       (/
        (fma (sqrt 2.0) (* t_0 (* (sin y) (- 1.0 (cos y)))) 2.0)
        (fma 1.5 (fma (cos x) t_2 (* (cos y) t_1)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -0.0059) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * -0.0625) * pow(sin(y), 2.0)))) / fma(1.5, fma(cos(y), t_1, (cos(x) * t_2)), 3.0);
	} else if (y <= 5.4e-5) {
		tmp = fma(sqrt(2.0), (t_0 * ((cos(x) + -1.0) * fma(-0.0625, sin(x), y))), 2.0) / fma(1.5, fma(cos(x), t_2, (cos(y) * (4.0 / (3.0 + sqrt(5.0))))), 3.0);
	} else {
		tmp = fma(sqrt(2.0), (t_0 * (sin(y) * (1.0 - cos(y)))), 2.0) / fma(1.5, fma(cos(x), t_2, (cos(y) * t_1)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -0.0059)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * -0.0625) * (sin(y) ^ 2.0)))) / fma(1.5, fma(cos(y), t_1, Float64(cos(x) * t_2)), 3.0));
	elseif (y <= 5.4e-5)
		tmp = Float64(fma(sqrt(2.0), Float64(t_0 * Float64(Float64(cos(x) + -1.0) * fma(-0.0625, sin(x), y))), 2.0) / fma(1.5, fma(cos(x), t_2, Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0))))), 3.0));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(t_0 * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / fma(1.5, fma(cos(x), t_2, Float64(cos(y) * t_1)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -0.0059], 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[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-5], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00589999999999999986

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if -0.00589999999999999986 < y < 5.3999999999999998e-5

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-+.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-sin.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if 5.3999999999999998e-5 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.7

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

   10. Applied rewrites 62.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0059:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x + -1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sin y, -0.0625, \sin x\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} + -1\\ t_3 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot t\_1\right), 3\right)\\ \mathbf{if}\;y \leq -0.0059:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_0 \cdot \left(\left(\cos x + -1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)\right), 2\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t\_0 \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sin y) -0.0625 (sin x)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (fma 1.5 (fma (cos x) t_2 (* (cos y) t_1)) 3.0)))
   (if (<= y -0.0059)
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (* (sqrt 2.0) -0.0625) (pow (sin y) 2.0))))
      (fma 1.5 (fma (cos y) t_1 (* (cos x) t_2)) 3.0))
     (if (<= y 5.4e-5)
       (/
        (fma
         (sqrt 2.0)
         (* t_0 (* (+ (cos x) -1.0) (fma -0.0625 (sin x) y)))
         2.0)
        t_3)
       (/ (fma (sqrt 2.0) (* t_0 (* (sin y) (- 1.0 (cos y)))) 2.0) t_3)))))

C

double code(double x, double y) {
	double t_0 = fma(sin(y), -0.0625, sin(x));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = fma(1.5, fma(cos(x), t_2, (cos(y) * t_1)), 3.0);
	double tmp;
	if (y <= -0.0059) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * -0.0625) * pow(sin(y), 2.0)))) / fma(1.5, fma(cos(y), t_1, (cos(x) * t_2)), 3.0);
	} else if (y <= 5.4e-5) {
		tmp = fma(sqrt(2.0), (t_0 * ((cos(x) + -1.0) * fma(-0.0625, sin(x), y))), 2.0) / t_3;
	} else {
		tmp = fma(sqrt(2.0), (t_0 * (sin(y) * (1.0 - cos(y)))), 2.0) / t_3;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sin(y), -0.0625, sin(x))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = fma(1.5, fma(cos(x), t_2, Float64(cos(y) * t_1)), 3.0)
	tmp = 0.0
	if (y <= -0.0059)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * -0.0625) * (sin(y) ^ 2.0)))) / fma(1.5, fma(cos(y), t_1, Float64(cos(x) * t_2)), 3.0));
	elseif (y <= 5.4e-5)
		tmp = Float64(fma(sqrt(2.0), Float64(t_0 * Float64(Float64(cos(x) + -1.0) * fma(-0.0625, sin(x), y))), 2.0) / t_3);
	else
		tmp = Float64(fma(sqrt(2.0), Float64(t_0 * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / t_3);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[y, -0.0059], 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[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-5], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.00589999999999999986

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if -0.00589999999999999986 < y < 5.3999999999999998e-5

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 99.5

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

   10. Applied rewrites 99.5%

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

   if 5.3999999999999998e-5 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.7

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

   10. Applied rewrites 62.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0059:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\left(\cos x + -1\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_1\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_1, \cos y \cdot t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (<= y -1.8e-5)
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (* (sqrt 2.0) -0.0625) (pow (sin y) 2.0))))
      (fma 1.5 (fma (cos y) t_0 (* (cos x) t_1)) 3.0))
     (if (<= y 8.2e-6)
       (/
        (fma
         0.3333333333333333
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma t_1 (cos x) t_0) 1.0))
       (/
        (fma
         (sqrt 2.0)
         (* (fma (sin y) -0.0625 (sin x)) (* (sin y) (- 1.0 (cos y))))
         2.0)
        (fma 1.5 (fma (cos x) t_1 (* (cos y) t_0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -1.8e-5) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * -0.0625) * pow(sin(y), 2.0)))) / fma(1.5, fma(cos(y), t_0, (cos(x) * t_1)), 3.0);
	} else if (y <= 8.2e-6) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (pow(sin(x), 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), t_0), 1.0);
	} else {
		tmp = fma(sqrt(2.0), (fma(sin(y), -0.0625, sin(x)) * (sin(y) * (1.0 - cos(y)))), 2.0) / fma(1.5, fma(cos(x), t_1, (cos(y) * t_0)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -1.8e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * -0.0625) * (sin(y) ^ 2.0)))) / fma(1.5, fma(cos(y), t_0, Float64(cos(x) * t_1)), 3.0));
	elseif (y <= 8.2e-6)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), t_0), 1.0));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(fma(sin(y), -0.0625, sin(x)) * Float64(sin(y) * Float64(1.0 - cos(y)))), 2.0) / fma(1.5, fma(cos(x), t_1, Float64(cos(y) * t_0)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -1.8e-5], 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[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-6], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.80000000000000005e-5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if -1.80000000000000005e-5 < y < 8.1999999999999994e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 98.9%

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

   if 8.1999999999999994e-6 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 62.7

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

   10. Applied rewrites 62.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\sin y \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot t\_0\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, \cos x \cdot t\_2\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_0, -0.0625 \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_2, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= y -1.8e-5)
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* (* (sqrt 2.0) -0.0625) t_0)))
      (fma 1.5 (fma (cos y) t_1 (* (cos x) t_2)) 3.0))
     (if (<= y 8.2e-6)
       (/
        (fma
         0.3333333333333333
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma t_2 (cos x) t_1) 1.0))
       (/
        (fma (* (sqrt 2.0) t_0) (* -0.0625 (- 1.0 (cos y))) 2.0)
        (fma
         1.5
         (fma (cos x) t_2 (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))))
         3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -1.8e-5) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * -0.0625) * t_0))) / fma(1.5, fma(cos(y), t_1, (cos(x) * t_2)), 3.0);
	} else if (y <= 8.2e-6) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (pow(sin(x), 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_2, cos(x), t_1), 1.0);
	} else {
		tmp = fma((sqrt(2.0) * t_0), (-0.0625 * (1.0 - cos(y))), 2.0) / fma(1.5, fma(cos(x), t_2, (cos(y) * (4.0 / (3.0 + sqrt(5.0))))), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -1.8e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * -0.0625) * t_0))) / fma(1.5, fma(cos(y), t_1, Float64(cos(x) * t_2)), 3.0));
	elseif (y <= 8.2e-6)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_2, cos(x), t_1), 1.0));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_0), Float64(-0.0625 * Float64(1.0 - cos(y))), 2.0) / fma(1.5, fma(cos(x), t_2, Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0))))), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -1.8e-5], 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] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-6], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.80000000000000005e-5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 51.0

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

   8. Applied rewrites 51.0%

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

   if -1.80000000000000005e-5 < y < 8.1999999999999994e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 98.9%

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

   if 8.1999999999999994e-6 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.0%

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower--.f64 N/A

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

       13. lower-cos.f64 62.1

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

   12. Applied rewrites 62.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot {\sin y}^{2}\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin y}^{2}, -0.0625 \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{3 + \sqrt{5}}\\ t_1 := {\sin y}^{2}\\ t_2 := -0.0625 \cdot \left(1 - \cos y\right)\\ t_3 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_3\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_3, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_1, t\_2, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_3, \cos y \cdot t\_0\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 4.0 (+ 3.0 (sqrt 5.0))))
        (t_1 (pow (sin y) 2.0))
        (t_2 (* -0.0625 (- 1.0 (cos y))))
        (t_3 (+ (sqrt 5.0) -1.0)))
   (if (<= y -1.8e-5)
     (/
      (fma t_1 (* (sqrt 2.0) t_2) 2.0)
      (fma 1.5 (fma (cos y) t_0 (* (cos x) t_3)) 3.0))
     (if (<= y 8.2e-6)
       (/
        (fma
         0.3333333333333333
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma t_3 (cos x) (- 3.0 (sqrt 5.0))) 1.0))
       (/
        (fma (* (sqrt 2.0) t_1) t_2 2.0)
        (fma 1.5 (fma (cos x) t_3 (* (cos y) t_0)) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 4.0 / (3.0 + sqrt(5.0));
	double t_1 = pow(sin(y), 2.0);
	double t_2 = -0.0625 * (1.0 - cos(y));
	double t_3 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -1.8e-5) {
		tmp = fma(t_1, (sqrt(2.0) * t_2), 2.0) / fma(1.5, fma(cos(y), t_0, (cos(x) * t_3)), 3.0);
	} else if (y <= 8.2e-6) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (pow(sin(x), 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_3, cos(x), (3.0 - sqrt(5.0))), 1.0);
	} else {
		tmp = fma((sqrt(2.0) * t_1), t_2, 2.0) / fma(1.5, fma(cos(x), t_3, (cos(y) * t_0)), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(4.0 / Float64(3.0 + sqrt(5.0)))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(-0.0625 * Float64(1.0 - cos(y)))
	t_3 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -1.8e-5)
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * t_2), 2.0) / fma(1.5, fma(cos(y), t_0, Float64(cos(x) * t_3)), 3.0));
	elseif (y <= 8.2e-6)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_3, cos(x), Float64(3.0 - sqrt(5.0))), 1.0));
	else
		tmp = Float64(fma(Float64(sqrt(2.0) * t_1), t_2, 2.0) / fma(1.5, fma(cos(x), t_3, Float64(cos(y) * t_0)), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -1.8e-5], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-6], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$3 * N[Cos[x], $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2 + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$3 + N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.80000000000000005e-5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 51.0

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

   10. Applied rewrites 51.0%

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

   if -1.80000000000000005e-5 < y < 8.1999999999999994e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 98.9%

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

   if 8.1999999999999994e-6 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.0%

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-sqrt.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower--.f64 N/A

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

       13. lower-cos.f64 62.1

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

   12. Applied rewrites 62.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot {\sin y}^{2}, -0.0625 \cdot \left(1 - \cos y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin y}^{2}\\ t_2 := \sqrt{5} + -1\\ t_3 := \cos x \cdot t\_2\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, t\_3\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_2, \cos x, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, t\_3\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin y) 2.0))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (* (cos x) t_2)))
   (if (<= y -1.8e-5)
     (/
      (fma t_1 (* (sqrt 2.0) (* -0.0625 (- 1.0 (cos y)))) 2.0)
      (fma 1.5 (fma (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))) t_3) 3.0))
     (if (<= y 8.2e-6)
       (/
        (fma
         0.3333333333333333
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma t_2 (cos x) t_0) 1.0))
       (/
        (fma t_1 (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) t_0 t_3) 3.0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = cos(x) * t_2;
	double tmp;
	if (y <= -1.8e-5) {
		tmp = fma(t_1, (sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (3.0 + sqrt(5.0))), t_3), 3.0);
	} else if (y <= 8.2e-6) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (pow(sin(x), 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_2, cos(x), t_0), 1.0);
	} else {
		tmp = fma(t_1, (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, t_3), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = Float64(cos(x) * t_2)
	tmp = 0.0
	if (y <= -1.8e-5)
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(3.0 + sqrt(5.0))), t_3), 3.0));
	elseif (y <= 8.2e-6)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_2, cos(x), t_0), 1.0));
	else
		tmp = Float64(fma(t_1, Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, t_3), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[y, -1.8e-5], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-6], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.80000000000000005e-5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\color{blue}{3 + \sqrt{5}}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \color{blue}{\left(1 - \cos y\right)}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

      13. lower-cos.f64 51.0

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \color{blue}{\cos y}\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 51.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)} \]

   if -1.80000000000000005e-5 < y < 8.1999999999999994e-6

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   12. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}} \]

   if 8.1999999999999994e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 62.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{3 + \sqrt{5}}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, \cos x \cdot t\_1\right), 3\right)}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (/
          (fma
           (pow (sin y) 2.0)
           (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625))
           2.0)
          (fma 1.5 (fma (cos y) t_0 (* (cos x) t_1)) 3.0))))
   (if (<= y -1.8e-5)
     t_2
     (if (<= y 8.2e-6)
       (/
        (fma
         0.3333333333333333
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma t_1 (cos x) t_0) 1.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, (cos(x) * t_1)), 3.0);
	double tmp;
	if (y <= -1.8e-5) {
		tmp = t_2;
	} else if (y <= 8.2e-6) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (pow(sin(x), 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), t_0), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, Float64(cos(x) * t_1)), 3.0))
	tmp = 0.0
	if (y <= -1.8e-5)
		tmp = t_2;
	elseif (y <= 8.2e-6)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), t_0), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e-5], t$95$2, If[LessEqual[y, 8.2e-6], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000005e-5 or 8.1999999999999994e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. Applied rewrites 56.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)}} \]

   if -1.80000000000000005e-5 < y < 8.1999999999999994e-6

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   12. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_1, \cos x, t\_0\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (/
          (fma
           (- 0.5 (* 0.5 (cos (+ y y))))
           (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625))
           2.0)
          (*
           3.0
           (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) (* (cos y) (/ t_0 2.0)))))))
   (if (<= y -1.8e-5)
     t_2
     (if (<= y 8.2e-6)
       (/
        (fma
         0.3333333333333333
         (* (sqrt 2.0) (* (pow (sin x) 2.0) (fma (cos x) -0.0625 0.0625)))
         0.6666666666666666)
        (fma 0.5 (fma t_1 (cos x) t_0) 1.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = fma((0.5 - (0.5 * cos((y + y)))), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	double tmp;
	if (y <= -1.8e-5) {
		tmp = t_2;
	} else if (y <= 8.2e-6) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (pow(sin(x), 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), t_0), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(y + y)))), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))))
	tmp = 0.0
	if (y <= -1.8e-5)
		tmp = t_2;
	elseif (y <= 8.2e-6)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * fma(cos(x), -0.0625, 0.0625))), 0.6666666666666666) / fma(0.5, fma(t_1, cos(x), t_0), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e-5], t$95$2, If[LessEqual[y, 8.2e-6], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000005e-5 or 8.1999999999999994e-6 < y

   1. Initial program 99.1%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 56.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 56.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -1.80000000000000005e-5 < y < 8.1999999999999994e-6

   1. Initial program 99.5%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)}} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   12. Applied rewrites 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(y + y\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := \frac{\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (/
          (fma
           (+ 0.5 (* -0.5 (cos (+ x x))))
           (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
           2.0)
          (*
           3.0
           (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) (* (cos y) (/ t_0 2.0)))))))
   (if (<= x -1.25e-5)
     t_2
     (if (<= x 7.5e-12)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) t_0 t_1) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = fma((0.5 + (-0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	double tmp;
	if (x <= -1.25e-5) {
		tmp = t_2;
	} else if (x <= 7.5e-12) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, t_1), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(fma(Float64(0.5 + Float64(-0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))))
	tmp = 0.0
	if (x <= -1.25e-5)
		tmp = t_2;
	elseif (x <= 7.5e-12)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_0, t_1), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 + N[(-0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e-5], t$95$2, If[LessEqual[x, 7.5e-12], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25000000000000006e-5 or 7.5e-12 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 60.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   if -1.25000000000000006e-5 < x < 7.5e-12

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right)\right), 3\right)} \]

      15. metadata-eval 99.1

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

   8. Applied rewrites 99.1%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 + -0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot t\_2\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 1\right)}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, \sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (fma (cos x) -0.0625 0.0625)))
   (if (<= x -0.2)
     (/
      (fma
       0.3333333333333333
       (* (sqrt 2.0) (* (pow (sin x) 2.0) t_2))
       0.6666666666666666)
      (fma 0.5 (fma t_0 (cos x) t_1) 1.0))
     (if (<= x 7.5e-12)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       (/
        (fma t_2 (* (sqrt 2.0) (- 0.5 (* 0.5 (cos (+ x x))))) 2.0)
        (fma 1.5 (fma (cos x) t_0 t_1) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(cos(x), -0.0625, 0.0625);
	double tmp;
	if (x <= -0.2) {
		tmp = fma(0.3333333333333333, (sqrt(2.0) * (pow(sin(x), 2.0) * t_2)), 0.6666666666666666) / fma(0.5, fma(t_0, cos(x), t_1), 1.0);
	} else if (x <= 7.5e-12) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = fma(t_2, (sqrt(2.0) * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0);
	}
	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 = fma(cos(x), -0.0625, 0.0625)
	tmp = 0.0
	if (x <= -0.2)
		tmp = Float64(fma(0.3333333333333333, Float64(sqrt(2.0) * Float64((sin(x) ^ 2.0) * t_2)), 0.6666666666666666) / fma(0.5, fma(t_0, cos(x), t_1), 1.0));
	elseif (x <= 7.5e-12)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = Float64(fma(t_2, Float64(sqrt(2.0) * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]}, If[LessEqual[x, -0.2], N[(N[(0.3333333333333333 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] / N[(0.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-12], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.20000000000000001

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\left(\frac{1}{2} \cdot 3\right)} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      7. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\left(\frac{1}{2} \cdot 3\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot 3, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \left(3 - \sqrt{5}\right)\right), 3\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \mathsf{fma}\left(\sin y, -0.0625, \sin x\right) \cdot \left(\mathsf{fma}\left(\sin x, -0.0625, \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \cos y \cdot \frac{4}{3 + \sqrt{5}}\right), 3\right)} \]

   10. Taylor expanded in y around 0

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}} \]

   12. Applied rewrites 63.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt{2} \cdot \left({\sin x}^{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right)\right), 0.6666666666666666\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} + -1, \cos x, 3 - \sqrt{5}\right), 1\right)}} \]

   if -0.20000000000000001 < x < 7.5e-12

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right)\right), 3\right)} \]

      15. metadata-eval 98.5

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]

   if 7.5e-12 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 57.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, 3 - \sqrt{5}\right), 3\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, 3 - \sqrt{5}\right), 3\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      15. lower-sqrt.f64 57.1

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 57.1%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.1%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \color{blue}{\sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, t\_0, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (fma (cos x) -0.0625 0.0625)
           (* (sqrt 2.0) (- 0.5 (* 0.5 (cos (+ x x)))))
           2.0)
          (fma 1.5 (fma (cos x) t_0 t_1) 3.0))))
   (if (<= x -0.2)
     t_2
     (if (<= x 7.5e-12)
       (/
        (fma (pow (sin y) 2.0) (* (sqrt 2.0) (fma (cos y) 0.0625 -0.0625)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma(fma(cos(x), -0.0625, 0.0625), (sqrt(2.0) * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0);
	double tmp;
	if (x <= -0.2) {
		tmp = t_2;
	} else if (x <= 7.5e-12) {
		tmp = fma(pow(sin(y), 2.0), (sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(fma(cos(x), -0.0625, 0.0625), Float64(sqrt(2.0) * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(1.5, fma(cos(x), t_0, t_1), 3.0))
	tmp = 0.0
	if (x <= -0.2)
		tmp = t_2;
	elseif (x <= 7.5e-12)
		tmp = Float64(fma((sin(y) ^ 2.0), Float64(sqrt(2.0) * fma(cos(y), 0.0625, -0.0625)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.2], t$95$2, If[LessEqual[x, 7.5e-12], N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[y], $MachinePrecision] * 0.0625 + -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.20000000000000001 or 7.5e-12 < x

   1. Initial program 99.0%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 61.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, 3 - \sqrt{5}\right), 3\right)} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, 3 - \sqrt{5}\right), 3\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

      15. lower-sqrt.f64 60.2

          \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

   8. Applied rewrites 60.2%

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.2%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \color{blue}{\sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   if -0.20000000000000001 < x < 7.5e-12

   1. Initial program 99.7%

      \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{\sin y}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{{\sin y}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \left(\frac{-1}{16} \cdot \left(1 - \cos y\right)\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. Applied rewrites 98.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 1\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1}} \]

      3. distribute-lft-out N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)\right)} + 3 \cdot 1} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right)} + 3 \cdot 1} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + 3 \cdot 1} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right)\right) + \color{blue}{3}} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos y \cdot \left(3 - \sqrt{5}\right) + \left(\sqrt{5} - 1\right), 3\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right)}, 3\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos y}, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \color{blue}{3 - \sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \color{blue}{\sqrt{5}}, \sqrt{5} - 1\right), 3\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}\right), 3\right)} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, \frac{1}{16}, \frac{-1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \color{blue}{\sqrt{5}} + \left(\mathsf{neg}\left(1\right)\right)\right), 3\right)} \]

      15. metadata-eval 98.5

          \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + \color{blue}{-1}\right), 3\right)} \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\mathsf{fma}\left({\sin y}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos y, 0.0625, -0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} + -1\right), 3\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 60.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (fma (cos x) -0.0625 0.0625)
   (* (sqrt 2.0) (- 0.5 (* 0.5 (cos (+ x x)))))
   2.0)
  (fma 1.5 (fma (cos x) (+ (sqrt 5.0) -1.0) (- 3.0 (sqrt 5.0))) 3.0)))

C

double code(double x, double y) {
	return fma(fma(cos(x), -0.0625, 0.0625), (sqrt(2.0) * (0.5 - (0.5 * cos((x + x))))), 2.0) / fma(1.5, fma(cos(x), (sqrt(5.0) + -1.0), (3.0 - sqrt(5.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(fma(cos(x), -0.0625, 0.0625), Float64(sqrt(2.0) * Float64(0.5 - Float64(0.5 * cos(Float64(x + x))))), 2.0) / fma(1.5, fma(cos(x), Float64(sqrt(5.0) + -1.0), Float64(3.0 - sqrt(5.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 66.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   15. lower-sqrt.f64 64.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 64.0%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), \color{blue}{\sqrt{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right)}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
11. Add Preprocessing

Alternative 25: 60.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (- 0.5 (* 0.5 (cos (+ x x))))
   (* (sqrt 2.0) (fma (cos x) -0.0625 0.0625))
   2.0)
  (fma 1.5 (- (fma (cos x) (+ (sqrt 5.0) -1.0) 3.0) (sqrt 5.0)) 3.0)))

C

double code(double x, double y) {
	return fma((0.5 - (0.5 * cos((x + x)))), (sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / fma(1.5, (fma(cos(x), (sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(fma(Float64(0.5 - Float64(0.5 * cos(Float64(x + x)))), Float64(sqrt(2.0) * fma(cos(x), -0.0625, 0.0625)), 2.0) / fma(1.5, Float64(fma(cos(x), Float64(sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 66.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   15. lower-sqrt.f64 64.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 64.0%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(x + x\right), \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]
11. Add Preprocessing

Alternative 26: 45.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   1.5
   (fma (cos y) (- 3.0 (sqrt 5.0)) (* (cos x) (+ (sqrt 5.0) -1.0)))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), (cos(x) * (sqrt(5.0) + -1.0))), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(cos(x) * Float64(sqrt(5.0) + -1.0))), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 66.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   15. lower-sqrt.f64 64.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 46.7%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]

12. Taylor expanded in x around inf

    \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 1\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + 3 \cdot 1}} \]

    3. distribute-lft-out N/A

       \[\leadsto \frac{2}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

    4. associate-*r* N/A

       \[\leadsto \frac{2}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

    6. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \cos y \cdot \left(3 - \sqrt{5}\right), 3\right)}} \]

14. Applied rewrites 49.0%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} + -1\right) \cdot \cos x\right), 3\right)}} \]

15. Final simplification 49.0%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \cos x \cdot \left(\sqrt{5} + -1\right)\right), 3\right)} \]
16. Add Preprocessing

Alternative 27: 43.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ 2.0 (fma 1.5 (- (fma (cos x) (+ (sqrt 5.0) -1.0) 3.0) (sqrt 5.0)) 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(1.5, (fma(cos(x), (sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(1.5, Float64(fma(cos(x), Float64(sqrt(5.0) + -1.0), 3.0) - sqrt(5.0)), 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + 3.0), $MachinePrecision] - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) \cdot \frac{-1}{16}} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot \frac{-1}{16}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\left(\cos x - 1\right) \cdot \frac{-1}{16}\right)\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right)}\right) + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{{\sin x}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}\right)} + 2}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \left(\frac{-1}{16} \cdot \left(\cos x - 1\right)\right) \cdot \sqrt{2}, 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

5. Applied rewrites 66.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(1 + \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 1\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1}} \]

   3. distribute-lft-out N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{3 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)\right)} + 3 \cdot 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\left(3 \cdot \frac{1}{2}\right) \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right)} + 3 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\frac{3}{2}} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + 3 \cdot 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\frac{3}{2} \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right)\right) + \color{blue}{3}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\color{blue}{\mathsf{fma}\left(\frac{3}{2}, \cos x \cdot \left(\sqrt{5} - 1\right) + \left(3 - \sqrt{5}\right), 3\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{5} - 1, 3 - \sqrt{5}\right)}, 3\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\color{blue}{\cos x}, \sqrt{5} - 1, 3 - \sqrt{5}\right), 3\right)} \]

   10. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + \left(\mathsf{neg}\left(1\right)\right)}, 3 - \sqrt{5}\right), 3\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + \color{blue}{-1}, 3 - \sqrt{5}\right), 3\right)} \]

   12. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} + -1}, 3 - \sqrt{5}\right), 3\right)} \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5}} + -1, 3 - \sqrt{5}\right), 3\right)} \]

   14. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, \frac{-1}{16}, \frac{1}{16}\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, \color{blue}{3 - \sqrt{5}}\right), 3\right)} \]

   15. lower-sqrt.f64 64.0

       \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \color{blue}{\sqrt{5}}\right), 3\right)} \]

8. Applied rewrites 64.0%

   \[\leadsto \frac{\mathsf{fma}\left({\sin x}^{2}, \sqrt{2} \cdot \mathsf{fma}\left(\cos x, -0.0625, 0.0625\right), 2\right)}{\color{blue}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
10. Step-by-step derivation

11. Applied rewrites 46.7%

    \[\leadsto \frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3 - \sqrt{5}\right), 3\right)} \]
12. Step-by-step derivation

13. Applied rewrites 46.7%

    \[\leadsto \color{blue}{\frac{2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos x, \sqrt{5} + -1, 3\right) - \sqrt{5}, 3\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))))