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

Percentage Accurate: 99.3% → 99.4%

Time: 21.9s

Alternatives: 31

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ t_2 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot t\_0, \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_1\right), 1\right)}\\ \mathbf{if}\;y \leq -0.3:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.64:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right) \cdot \sqrt{2}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* (- (sqrt 5.0) 1.0) (cos x)))
        (t_2
         (*
          0.3333333333333333
          (/
           (fma
            (* (sqrt 2.0) t_0)
            (* (- (sin x) (* 0.0625 (sin y))) (sin y))
            2.0)
           (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_1) 1.0)))))
   (if (<= y -0.3)
     t_2
     (if (<= y 0.64)
       (/
        (+
         (*
          (*
           (- (sin y) (/ (sin x) 16.0))
           (*
            (fma
             (fma
              (fma -0.0005208333333333333 (* y y) 0.010416666666666666)
              (* y y)
              -0.0625)
             y
             (sin x))
            (sqrt 2.0)))
          t_0)
         2.0)
        (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_1) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = (sqrt(5.0) - 1.0) * cos(x);
	double t_2 = 0.3333333333333333 * (fma((sqrt(2.0) * t_0), ((sin(x) - (0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), t_1), 1.0));
	double tmp;
	if (y <= -0.3) {
		tmp = t_2;
	} else if (y <= 0.64) {
		tmp = ((((sin(y) - (sin(x) / 16.0)) * (fma(fma(fma(-0.0005208333333333333, (y * y), 0.010416666666666666), (y * y), -0.0625), y, sin(x)) * sqrt(2.0))) * t_0) + 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_1), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) * cos(x))
	t_2 = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * t_0), Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_1), 1.0)))
	tmp = 0.0
	if (y <= -0.3)
		tmp = t_2;
	elseif (y <= 0.64)
		tmp = Float64(Float64(Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(fma(fma(fma(-0.0005208333333333333, Float64(y * y), 0.010416666666666666), Float64(y * y), -0.0625), y, sin(x)) * sqrt(2.0))) * t_0) + 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_1), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.3], t$95$2, If[LessEqual[y, 0.64], N[(N[(N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.0005208333333333333 * N[(y * y), $MachinePrecision] + 0.010416666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.299999999999999989 or 0.640000000000000013 < 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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

   if -0.299999999999999989 < y < 0.640000000000000013

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 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%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(\sin x + y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right)\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) + \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}\right) \cdot y} + \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) - \frac{1}{16}, y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right), y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\frac{-1}{16}}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{96} + \frac{-1}{1920} \cdot {y}^{2}, {y}^{2}, \frac{-1}{16}\right)}, y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1920} \cdot {y}^{2} + \frac{1}{96}}, {y}^{2}, \frac{-1}{16}\right), y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1920}, {y}^{2}, \frac{1}{96}\right)}, {y}^{2}, \frac{-1}{16}\right), y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, \color{blue}{y \cdot y}, \frac{1}{96}\right), {y}^{2}, \frac{-1}{16}\right), y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, \color{blue}{y \cdot y}, \frac{1}{96}\right), {y}^{2}, \frac{-1}{16}\right), y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, y \cdot y, \frac{1}{96}\right), \color{blue}{y \cdot y}, \frac{-1}{16}\right), y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1920}, y \cdot y, \frac{1}{96}\right), \color{blue}{y \cdot y}, \frac{-1}{16}\right), y, \sin x\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

      14. lower-sin.f64 99.7

          \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \color{blue}{\sin x}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

   10. Applied rewrites 99.7%

       \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.3:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;y \leq 0.64:\\ \;\;\;\;\frac{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005208333333333333, y \cdot y, 0.010416666666666666\right), y \cdot y, -0.0625\right), y, \sin x\right) \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ t_1 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0\right), 1\right)}\\ \mathbf{if}\;y \leq -0.27:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right), y \cdot y, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (sqrt 5.0) 1.0) (cos x)))
        (t_1
         (*
          0.3333333333333333
          (/
           (fma
            (* (sqrt 2.0) (- (cos x) (cos y)))
            (* (- (sin x) (* 0.0625 (sin y))) (sin y))
            2.0)
           (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_0) 1.0)))))
   (if (<= y -0.27)
     t_1
     (if (<= y 0.62)
       (/
        (fma
         (fma -0.0625 (sin y) (sin x))
         (*
          (*
           (fma
            (fma -0.041666666666666664 (* y y) 0.5)
            (* y y)
            (- (cos x) 1.0))
           (sqrt 2.0))
          (fma -0.0625 (sin x) (sin y)))
         2.0)
        (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_0) 3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) - 1.0) * cos(x);
	double t_1 = 0.3333333333333333 * (fma((sqrt(2.0) * (cos(x) - cos(y))), ((sin(x) - (0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), t_0), 1.0));
	double tmp;
	if (y <= -0.27) {
		tmp = t_1;
	} else if (y <= 0.62) {
		tmp = fma(fma(-0.0625, sin(y), sin(x)), ((fma(fma(-0.041666666666666664, (y * y), 0.5), (y * y), (cos(x) - 1.0)) * sqrt(2.0)) * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_0), 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) - 1.0) * cos(x))
	t_1 = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_0), 1.0)))
	tmp = 0.0
	if (y <= -0.27)
		tmp = t_1;
	elseif (y <= 0.62)
		tmp = Float64(fma(fma(-0.0625, sin(y), sin(x)), Float64(Float64(fma(fma(-0.041666666666666664, Float64(y * y), 0.5), Float64(y * y), Float64(cos(x) - 1.0)) * sqrt(2.0)) * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_0), 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.27], t$95$1, If[LessEqual[y, 0.62], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(-0.041666666666666664 * N[(y * y), $MachinePrecision] + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.27000000000000002 or 0.619999999999999996 < 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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

   if -0.27000000000000002 < y < 0.619999999999999996

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 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%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\left(\left(\cos x + {y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right)\right) - 1\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]
   11. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\left(\left(\cos x + {y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       2. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) + \cos x\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\left({y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) + \cos x\right) + \color{blue}{-1}\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       4. associate-+l+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) + \left(\cos x + -1\right)\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot {y}^{2}} + \left(\cos x + -1\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot {y}^{2} + \left(\cos x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       7. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}\right) \cdot {y}^{2} + \color{blue}{\left(\cos x - 1\right)}\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{24} \cdot {y}^{2}, {y}^{2}, \cos x - 1\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       9. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{24} \cdot {y}^{2} + \frac{1}{2}}, {y}^{2}, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       10. lower-fma.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{24}, {y}^{2}, \frac{1}{2}\right)}, {y}^{2}, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       11. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, \color{blue}{y \cdot y}, \frac{1}{2}\right), {y}^{2}, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, \color{blue}{y \cdot y}, \frac{1}{2}\right), {y}^{2}, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       13. unpow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, y \cdot y, \frac{1}{2}\right), \color{blue}{y \cdot y}, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, y \cdot y, \frac{1}{2}\right), \color{blue}{y \cdot y}, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       15. lower--.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{16}, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{24}, y \cdot y, \frac{1}{2}\right), y \cdot y, \color{blue}{\cos x - 1}\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       16. lower-cos.f64 99.7

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right), y \cdot y, \color{blue}{\cos x} - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right), y \cdot y, \cos x - 1\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.27:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;y \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, y \cdot y, 0.5\right), y \cdot y, \cos x - 1\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \left(\cos x - \cos y\right)\\ t_1 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ t_2 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(t\_0, \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_1\right), 1\right)}\\ \mathbf{if}\;y \leq -0.23:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sin x\right), t\_0 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (- (cos x) (cos y))))
        (t_1 (* (- (sqrt 5.0) 1.0) (cos x)))
        (t_2
         (*
          0.3333333333333333
          (/
           (fma t_0 (* (- (sin x) (* 0.0625 (sin y))) (sin y)) 2.0)
           (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_1) 1.0)))))
   (if (<= y -0.23)
     t_2
     (if (<= y 0.62)
       (/
        (fma
         (fma (fma 0.010416666666666666 (* y y) -0.0625) y (sin x))
         (* t_0 (fma -0.0625 (sin x) (sin y)))
         2.0)
        (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_1) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * (cos(x) - cos(y));
	double t_1 = (sqrt(5.0) - 1.0) * cos(x);
	double t_2 = 0.3333333333333333 * (fma(t_0, ((sin(x) - (0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), t_1), 1.0));
	double tmp;
	if (y <= -0.23) {
		tmp = t_2;
	} else if (y <= 0.62) {
		tmp = fma(fma(fma(0.010416666666666666, (y * y), -0.0625), y, sin(x)), (t_0 * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_1), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * Float64(cos(x) - cos(y)))
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) * cos(x))
	t_2 = Float64(0.3333333333333333 * Float64(fma(t_0, Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_1), 1.0)))
	tmp = 0.0
	if (y <= -0.23)
		tmp = t_2;
	elseif (y <= 0.62)
		tmp = Float64(fma(fma(fma(0.010416666666666666, Float64(y * y), -0.0625), y, sin(x)), Float64(t_0 * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_1), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(t$95$0 * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.23], t$95$2, If[LessEqual[y, 0.62], N[(N[(N[(N[(0.010416666666666666 * N[(y * y), $MachinePrecision] + -0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.23000000000000001 or 0.619999999999999996 < 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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

   if -0.23000000000000001 < y < 0.619999999999999996

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 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%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in y around 0

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sin x + y \cdot \left(\frac{1}{96} \cdot {y}^{2} - \frac{1}{16}\right)}, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{1}{96} \cdot {y}^{2} - \frac{1}{16}\right) + \sin x}, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{96} \cdot {y}^{2} - \frac{1}{16}\right) \cdot y} + \sin x, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{96} \cdot {y}^{2} - \frac{1}{16}, y, \sin x\right)}, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       4. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{96} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{16}\right)\right)}, y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       5. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{96} \cdot {y}^{2} + \color{blue}{\frac{-1}{16}}, y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{96}, {y}^{2}, \frac{-1}{16}\right)}, y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       7. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{96}, \color{blue}{y \cdot y}, \frac{-1}{16}\right), y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{96}, \color{blue}{y \cdot y}, \frac{-1}{16}\right), y, \sin x\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(\frac{-1}{16}, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(\frac{3}{2}, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

       9. lower-sin.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \color{blue}{\sin x}\right), \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]

   12. Applied rewrites 99.7%

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sin x\right)}, \left(\left(\cos x - \cos y\right) \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.23:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;y \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010416666666666666, y \cdot y, -0.0625\right), y, \sin x\right), \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)\\ t_1 := \sqrt{2} \cdot \left(\cos x - \cos y\right)\\ t_2 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(t\_1, \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.23:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          0.5
          (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
          1.0))
        (t_1 (* (sqrt 2.0) (- (cos x) (cos y))))
        (t_2
         (*
          0.3333333333333333
          (/ (fma t_1 (* (- (sin x) (* 0.0625 (sin y))) (sin y)) 2.0) t_0))))
   (if (<= y -0.23)
     t_2
     (if (<= y 0.62)
       (*
        (/
         (fma
          t_1
          (*
           (fma (fma (* y y) 0.010416666666666666 -0.0625) y (sin x))
           (- (sin y) (* 0.0625 (sin x))))
          2.0)
         t_0)
        0.3333333333333333)
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 1.0);
	double t_1 = sqrt(2.0) * (cos(x) - cos(y));
	double t_2 = 0.3333333333333333 * (fma(t_1, ((sin(x) - (0.0625 * sin(y))) * sin(y)), 2.0) / t_0);
	double tmp;
	if (y <= -0.23) {
		tmp = t_2;
	} else if (y <= 0.62) {
		tmp = (fma(t_1, (fma(fma((y * y), 0.010416666666666666, -0.0625), y, sin(x)) * (sin(y) - (0.0625 * sin(x)))), 2.0) / t_0) * 0.3333333333333333;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 1.0)
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) - cos(y)))
	t_2 = Float64(0.3333333333333333 * Float64(fma(t_1, Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)), 2.0) / t_0))
	tmp = 0.0
	if (y <= -0.23)
		tmp = t_2;
	elseif (y <= 0.62)
		tmp = Float64(Float64(fma(t_1, Float64(fma(fma(Float64(y * y), 0.010416666666666666, -0.0625), y, sin(x)) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / t_0) * 0.3333333333333333);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.23], t$95$2, If[LessEqual[y, 0.62], N[(N[(N[(t$95$1 * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.010416666666666666 + -0.0625), $MachinePrecision] * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.23000000000000001 or 0.619999999999999996 < 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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

   if -0.23000000000000001 < y < 0.619999999999999996

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 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%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - \frac{1}{16} \cdot \sin x\right) \cdot \left(\sin x + y \cdot \left(\frac{1}{96} \cdot {y}^{2} - \frac{1}{16}\right)\right), 2\right)}{\mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot \frac{1}{3} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin y - 0.0625 \cdot \sin x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.23:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;y \leq 0.62:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.010416666666666666, -0.0625\right), y, \sin x\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot \left(\cos x - \cos y\right)\\ t_1 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ t_2 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(t\_0, \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_1\right), 1\right)}\\ \mathbf{if}\;y \leq -0.056:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.053:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, y, \sin x\right), t\_0 \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) (- (cos x) (cos y))))
        (t_1 (* (- (sqrt 5.0) 1.0) (cos x)))
        (t_2
         (*
          0.3333333333333333
          (/
           (fma t_0 (* (- (sin x) (* 0.0625 (sin y))) (sin y)) 2.0)
           (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_1) 1.0)))))
   (if (<= y -0.056)
     t_2
     (if (<= y 0.053)
       (/
        (fma (fma -0.0625 y (sin x)) (* t_0 (fma -0.0625 (sin x) (sin y))) 2.0)
        (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_1) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * (cos(x) - cos(y));
	double t_1 = (sqrt(5.0) - 1.0) * cos(x);
	double t_2 = 0.3333333333333333 * (fma(t_0, ((sin(x) - (0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), t_1), 1.0));
	double tmp;
	if (y <= -0.056) {
		tmp = t_2;
	} else if (y <= 0.053) {
		tmp = fma(fma(-0.0625, y, sin(x)), (t_0 * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_1), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * Float64(cos(x) - cos(y)))
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) * cos(x))
	t_2 = Float64(0.3333333333333333 * Float64(fma(t_0, Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_1), 1.0)))
	tmp = 0.0
	if (y <= -0.056)
		tmp = t_2;
	elseif (y <= 0.053)
		tmp = Float64(fma(fma(-0.0625, y, sin(x)), Float64(t_0 * fma(-0.0625, sin(x), sin(y))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_1), 3.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(t$95$0 * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.056], t$95$2, If[LessEqual[y, 0.053], N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(-0.0625 * N[Sin[x], $MachinePrecision] + N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0560000000000000012 or 0.0529999999999999985 < 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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

   if -0.0560000000000000012 < y < 0.0529999999999999985

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 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%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-sin.f64 99.5

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

   12. Applied rewrites 99.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.056:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;y \leq 0.053:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, y, \sin x\right), \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \mathsf{fma}\left(-0.0625, \sin x, \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)\\ t_1 := \sqrt{2} \cdot \left(\cos x - \cos y\right)\\ t_2 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(t\_1, \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{t\_0}\\ \mathbf{if}\;y \leq -0.056:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.053:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{t\_0} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          0.5
          (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
          1.0))
        (t_1 (* (sqrt 2.0) (- (cos x) (cos y))))
        (t_2
         (*
          0.3333333333333333
          (/ (fma t_1 (* (- (sin x) (* 0.0625 (sin y))) (sin y)) 2.0) t_0))))
   (if (<= y -0.056)
     t_2
     (if (<= y 0.053)
       (*
        (/
         (fma
          t_1
          (* (fma y -0.0625 (sin x)) (- (sin y) (* 0.0625 (sin x))))
          2.0)
         t_0)
        0.3333333333333333)
       t_2))))

C

double code(double x, double y) {
	double t_0 = fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 1.0);
	double t_1 = sqrt(2.0) * (cos(x) - cos(y));
	double t_2 = 0.3333333333333333 * (fma(t_1, ((sin(x) - (0.0625 * sin(y))) * sin(y)), 2.0) / t_0);
	double tmp;
	if (y <= -0.056) {
		tmp = t_2;
	} else if (y <= 0.053) {
		tmp = (fma(t_1, (fma(y, -0.0625, sin(x)) * (sin(y) - (0.0625 * sin(x)))), 2.0) / t_0) * 0.3333333333333333;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 1.0)
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) - cos(y)))
	t_2 = Float64(0.3333333333333333 * Float64(fma(t_1, Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)), 2.0) / t_0))
	tmp = 0.0
	if (y <= -0.056)
		tmp = t_2;
	elseif (y <= 0.053)
		tmp = Float64(Float64(fma(t_1, Float64(fma(y, -0.0625, sin(x)) * Float64(sin(y) - Float64(0.0625 * sin(x)))), 2.0) / t_0) * 0.3333333333333333);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.3333333333333333 * N[(N[(t$95$1 * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.056], t$95$2, If[LessEqual[y, 0.053], N[(N[(N[(t$95$1 * N[(N[(y * -0.0625 + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0560000000000000012 or 0.0529999999999999985 < 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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

   if -0.0560000000000000012 < y < 0.0529999999999999985

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 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%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 99.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.056:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;y \leq 0.053:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \mathsf{fma}\left(y, -0.0625, \sin x\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right), 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_0, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{if}\;y \leq -0.018:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.5, \cos x - 1\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          0.3333333333333333
          (/
           (fma
            (* (sqrt 2.0) (- (cos x) (cos y)))
            (* (- (sin x) (* 0.0625 (sin y))) (sin y))
            2.0)
           (fma 0.5 (fma (cos y) t_0 (* (- (sqrt 5.0) 1.0) (cos x))) 1.0)))))
   (if (<= y -0.018)
     t_1
     (if (<= y 0.0135)
       (/
        (+
         (*
          (fma (* y y) 0.5 (- (cos x) 1.0))
          (*
           (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0))))
         2.0)
        (*
         (fma
          t_0
          (fma (* -0.25 y) y 0.5)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       t_1))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 0.3333333333333333 * (fma((sqrt(2.0) * (cos(x) - cos(y))), ((sin(x) - (0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), t_0, ((sqrt(5.0) - 1.0) * cos(x))), 1.0));
	double tmp;
	if (y <= -0.018) {
		tmp = t_1;
	} else if (y <= 0.0135) {
		tmp = ((fma((y * y), 0.5, (cos(x) - 1.0)) * (((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0)))) + 2.0) / (fma(t_0, fma((-0.25 * y), y, 0.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(0.3333333333333333 * Float64(fma(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))), Float64(Float64(sin(x) - Float64(0.0625 * sin(y))) * sin(y)), 2.0) / fma(0.5, fma(cos(y), t_0, Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 1.0)))
	tmp = 0.0
	if (y <= -0.018)
		tmp = t_1;
	elseif (y <= 0.0135)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.5, Float64(cos(x) - 1.0)) * Float64(Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))) + 2.0) / Float64(fma(t_0, fma(Float64(-0.25 * y), y, 0.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.018], t$95$1, If[LessEqual[y, 0.0135], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.5 + N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[(N[(-0.25 * y), $MachinePrecision] * y + 0.5), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0179999999999999986 or 0.0134999999999999998 < 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 y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.8%

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

   if -0.0179999999999999986 < y < 0.0134999999999999998

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 99.3

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

   8. Applied rewrites 99.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.018:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.5, \cos x - 1\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(\sqrt{2} \cdot \left(\cos x - \cos y\right), \left(\sin x - 0.0625 \cdot \sin y\right) \cdot \sin y, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ \mathbf{if}\;y \leq -0.018:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_0, t\_1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.5, \cos x - 1\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_1\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (* (- (sqrt 5.0) 1.0) (cos x))))
   (if (<= y -0.018)
     (/
      (+
       (* (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)) (- (cos x) (cos y)))
       2.0)
      (fma 1.5 (fma (cos y) t_0 t_1) 3.0))
     (if (<= y 0.0135)
       (/
        (+
         (*
          (fma (* y y) 0.5 (- (cos x) 1.0))
          (*
           (* (- (sin x) (/ (sin y) 16.0)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0))))
         2.0)
        (*
         (fma
          t_0
          (fma (* -0.25 y) y 0.5)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       (/
        (fma
         (fma -0.0625 (sin y) (sin x))
         (* (- 1.0 (cos y)) (* (sqrt 2.0) (sin y)))
         2.0)
        (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_1) 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) * cos(x);
	double tmp;
	if (y <= -0.018) {
		tmp = ((((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(y), t_0, t_1), 3.0);
	} else if (y <= 0.0135) {
		tmp = ((fma((y * y), 0.5, (cos(x) - 1.0)) * (((sin(x) - (sin(y) / 16.0)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0)))) + 2.0) / (fma(t_0, fma((-0.25 * y), y, 0.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = fma(fma(-0.0625, sin(y), sin(x)), ((1.0 - cos(y)) * (sqrt(2.0) * sin(y))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_1), 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) * cos(x))
	tmp = 0.0
	if (y <= -0.018)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(y), t_0, t_1), 3.0));
	elseif (y <= 0.0135)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.5, Float64(cos(x) - 1.0)) * Float64(Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))) + 2.0) / Float64(fma(t_0, fma(Float64(-0.25 * y), y, 0.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(fma(fma(-0.0625, sin(y), sin(x)), Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * sin(y))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_1), 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.018], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0135], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.5 + N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$0 * N[(N[(-0.25 * y), $MachinePrecision] * y + 0.5), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0179999999999999986

   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 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. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 56.2

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

   8. Applied rewrites 56.2%

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

   if -0.0179999999999999986 < y < 0.0134999999999999998

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 99.3

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

   8. Applied rewrites 99.3%

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

   if 0.0134999999999999998 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 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%

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 65.3

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

   12. Applied rewrites 65.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.018:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.5, \cos x - 1\right) \cdot \left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) + 2}{\mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ \mathbf{if}\;y \leq -0.018:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_0 + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_2\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_2\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (- (sqrt 5.0) 1.0) (cos x))))
   (if (<= y -0.018)
     (/
      (+ (* (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)) t_0) 2.0)
      (fma 1.5 (fma (cos y) t_1 t_2) 3.0))
     (if (<= y 0.0135)
       (/
        (+
         (*
          (*
           (* (fma -0.0625 y (sin x)) (sqrt 2.0))
           (- (sin y) (/ (sin x) 16.0)))
          t_0)
         2.0)
        (*
         (fma
          t_1
          (fma (* -0.25 y) y 0.5)
          (fma (fma (sqrt 5.0) 0.5 -0.5) (cos x) 1.0))
         3.0))
       (/
        (fma
         (fma -0.0625 (sin y) (sin x))
         (* (- 1.0 (cos y)) (* (sqrt 2.0) (sin y)))
         2.0)
        (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_2) 3.0))))))

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 = (sqrt(5.0) - 1.0) * cos(x);
	double tmp;
	if (y <= -0.018) {
		tmp = ((((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / fma(1.5, fma(cos(y), t_1, t_2), 3.0);
	} else if (y <= 0.0135) {
		tmp = ((((fma(-0.0625, y, sin(x)) * sqrt(2.0)) * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / (fma(t_1, fma((-0.25 * y), y, 0.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0);
	} else {
		tmp = fma(fma(-0.0625, sin(y), sin(x)), ((1.0 - cos(y)) * (sqrt(2.0) * sin(y))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_2), 3.0);
	}
	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(Float64(sqrt(5.0) - 1.0) * cos(x))
	tmp = 0.0
	if (y <= -0.018)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / fma(1.5, fma(cos(y), t_1, t_2), 3.0));
	elseif (y <= 0.0135)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.0625, y, sin(x)) * sqrt(2.0)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / Float64(fma(t_1, fma(Float64(-0.25 * y), y, 0.5), fma(fma(sqrt(5.0), 0.5, -0.5), cos(x), 1.0)) * 3.0));
	else
		tmp = Float64(fma(fma(-0.0625, sin(y), sin(x)), Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * sin(y))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_2), 3.0));
	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[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.018], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0135], N[(N[(N[(N[(N[(N[(-0.0625 * y + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(t$95$1 * N[(N[(-0.25 * y), $MachinePrecision] * y + 0.5), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0179999999999999986

   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 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. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 56.2

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

   8. Applied rewrites 56.2%

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

   if -0.0179999999999999986 < y < 0.0134999999999999998

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. associate-+l+ N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 99.3

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

   8. Applied rewrites 99.3%

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

   if 0.0134999999999999998 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 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%

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 65.3

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

   12. Applied rewrites 65.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.018:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(-0.0625, y, \sin x\right) \cdot \sqrt{2}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(3 - \sqrt{5}, \mathsf{fma}\left(-0.25 \cdot y, y, 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), \cos x, 1\right)\right) \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0625, \sin y, \sin x\right)\\ t_1 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ t_2 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_1\right), 3\right)\\ \mathbf{if}\;y \leq -0.017:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_1\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0042:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\sin x, -0.0625, y\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma -0.0625 (sin y) (sin x)))
        (t_1 (* (- (sqrt 5.0) 1.0) (cos x)))
        (t_2 (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_1) 3.0)))
   (if (<= y -0.017)
     (/
      (+
       (* (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)) (- (cos x) (cos y)))
       2.0)
      (fma 1.5 (fma (cos y) (- 3.0 (sqrt 5.0)) t_1) 3.0))
     (if (<= y 0.0042)
       (/
        (fma
         t_0
         (* (fma (sin x) -0.0625 y) (* (- (cos x) 1.0) (sqrt 2.0)))
         2.0)
        t_2)
       (/ (fma t_0 (* (- 1.0 (cos y)) (* (sqrt 2.0) (sin y))) 2.0) t_2)))))

C

double code(double x, double y) {
	double t_0 = fma(-0.0625, sin(y), sin(x));
	double t_1 = (sqrt(5.0) - 1.0) * cos(x);
	double t_2 = fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_1), 3.0);
	double tmp;
	if (y <= -0.017) {
		tmp = ((((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0)) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), t_1), 3.0);
	} else if (y <= 0.0042) {
		tmp = fma(t_0, (fma(sin(x), -0.0625, y) * ((cos(x) - 1.0) * sqrt(2.0))), 2.0) / t_2;
	} else {
		tmp = fma(t_0, ((1.0 - cos(y)) * (sqrt(2.0) * sin(y))), 2.0) / t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(-0.0625, sin(y), sin(x))
	t_1 = Float64(Float64(sqrt(5.0) - 1.0) * cos(x))
	t_2 = fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_1), 3.0)
	tmp = 0.0
	if (y <= -0.017)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0)) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), t_1), 3.0));
	elseif (y <= 0.0042)
		tmp = Float64(fma(t_0, Float64(fma(sin(x), -0.0625, y) * Float64(Float64(cos(x) - 1.0) * sqrt(2.0))), 2.0) / t_2);
	else
		tmp = Float64(fma(t_0, Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * sin(y))), 2.0) / t_2);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[y, -0.017], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0042], N[(N[(t$95$0 * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625 + y), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$0 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.017000000000000001

   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 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. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 56.2

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

   8. Applied rewrites 56.2%

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

   if -0.017000000000000001 < y < 0.00419999999999999974

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 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%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in y around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-sqrt.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-sin.f64 99.2

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

   12. Applied rewrites 99.2%

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

   if 0.00419999999999999974 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 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%

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 65.3

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

   12. Applied rewrites 65.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.017:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;y \leq 0.0042:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \mathsf{fma}\left(\sin x, -0.0625, y\right) \cdot \left(\left(\cos x - 1\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{5} - 1\\ t_3 := t\_2 \cdot \cos x\\ \mathbf{if}\;y \leq -0.0072:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot t\_0 + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_3\right), 3\right)}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t\_0 + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_2, \cos x, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_3\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (- (sqrt 5.0) 1.0))
        (t_3 (* t_2 (cos x))))
   (if (<= y -0.0072)
     (/
      (+ (* (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)) t_0) 2.0)
      (fma 1.5 (fma (cos y) t_1 t_3) 3.0))
     (if (<= y 5.6e-6)
       (/
        (+ (* (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))) t_0) 2.0)
        (fma 1.5 (fma t_2 (cos x) t_1) 3.0))
       (/
        (fma
         (fma -0.0625 (sin y) (sin x))
         (* (- 1.0 (cos y)) (* (sqrt 2.0) (sin y)))
         2.0)
        (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_3) 3.0))))))

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 = sqrt(5.0) - 1.0;
	double t_3 = t_2 * cos(x);
	double tmp;
	if (y <= -0.0072) {
		tmp = ((((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / fma(1.5, fma(cos(y), t_1, t_3), 3.0);
	} else if (y <= 5.6e-6) {
		tmp = ((((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * t_0) + 2.0) / fma(1.5, fma(t_2, cos(x), t_1), 3.0);
	} else {
		tmp = fma(fma(-0.0625, sin(y), sin(x)), ((1.0 - cos(y)) * (sqrt(2.0) * sin(y))), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_3), 3.0);
	}
	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(sqrt(5.0) - 1.0)
	t_3 = Float64(t_2 * cos(x))
	tmp = 0.0
	if (y <= -0.0072)
		tmp = Float64(Float64(Float64(Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0)) * t_0) + 2.0) / fma(1.5, fma(cos(y), t_1, t_3), 3.0));
	elseif (y <= 5.6e-6)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_0) + 2.0) / fma(1.5, fma(t_2, cos(x), t_1), 3.0));
	else
		tmp = Float64(fma(fma(-0.0625, sin(y), sin(x)), Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * sin(y))), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_3), 3.0));
	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[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0072], N[(N[(N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e-6], N[(N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$2 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0625 * N[Sin[y], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0071999999999999998

   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 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. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sqrt.f64 56.2

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

   8. Applied rewrites 56.2%

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

   if -0.0071999999999999998 < y < 5.59999999999999975e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

   10. Applied rewrites 98.7%

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

   if 5.59999999999999975e-6 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 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%

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 65.3

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

   12. Applied rewrites 65.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0072:\\ \;\;\;\;\frac{\left(\left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \sin y, \sin x\right), \left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot \sin y\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(\sqrt{2} \cdot x, 1.00390625 \cdot \sin y, \left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, t\_1, \sqrt{5}\right), -0.5\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_0 \cdot \cos x\right), 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.000145)
     (/
      (+
       (*
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
        (- (cos x) (cos y)))
       2.0)
      (fma 1.5 (fma t_0 (cos x) t_1) 3.0))
     (if (<= x 0.00012)
       (/
        (fma
         (- 1.0 (cos y))
         (fma
          (* (sqrt 2.0) x)
          (* 1.00390625 (sin y))
          (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)))
         2.0)
        (fma (fma 0.5 (fma (cos y) t_1 (sqrt 5.0)) -0.5) 3.0 3.0))
       (/
        (fma (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0)) (pow (sin x) 2.0) 2.0)
        (fma
         1.5
         (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) (* t_0 (cos x)))
         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 tmp;
	if (x <= -0.000145) {
		tmp = ((((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y))) + 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	} else if (x <= 0.00012) {
		tmp = fma((1.0 - cos(y)), fma((sqrt(2.0) * x), (1.00390625 * sin(y)), ((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0, 3.0);
	} else {
		tmp = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), (t_0 * cos(x))), 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))
	tmp = 0.0
	if (x <= -0.000145)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) - cos(y))) + 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0));
	elseif (x <= 0.00012)
		tmp = Float64(fma(Float64(1.0 - cos(y)), fma(Float64(sqrt(2.0) * x), Float64(1.00390625 * sin(y)), Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(cos(y), t_1, sqrt(5.0)), -0.5), 3.0, 3.0));
	else
		tmp = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), Float64(t_0 * cos(x))), 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]}, If[LessEqual[x, -0.000145], N[(N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $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] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00012], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-4

   1. Initial program 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sin.f64 69.1

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

   7. Applied rewrites 69.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

   10. Applied rewrites 65.9%

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

   if -1.45e-4 < x < 1.20000000000000003e-4

   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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   if 1.20000000000000003e-4 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 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%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 54.4

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

   10. Applied rewrites 54.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;\frac{\left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(\sqrt{2} \cdot x, 1.00390625 \cdot \sin y, \left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{if}\;x \leq -0.000145:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - \cos y, \mathsf{fma}\left(\sqrt{2} \cdot x, 1.00390625 \cdot \sin y, \left({\sin y}^{2} \cdot -0.0625\right) \cdot \sqrt{2}\right), 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (/
          (fma
           (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma
           1.5
           (fma
            (cos y)
            (/ 4.0 (+ (sqrt 5.0) 3.0))
            (* (- (sqrt 5.0) 1.0) (cos x)))
           3.0))))
   (if (<= x -0.000145)
     t_0
     (if (<= x 0.00012)
       (/
        (fma
         (- 1.0 (cos y))
         (fma
          (* (sqrt 2.0) x)
          (* 1.00390625 (sin y))
          (* (* (pow (sin y) 2.0) -0.0625) (sqrt 2.0)))
         2.0)
        (fma
         (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -0.5)
         3.0
         3.0))
       t_0))))

C

double code(double x, double y) {
	double t_0 = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	double tmp;
	if (x <= -0.000145) {
		tmp = t_0;
	} else if (x <= 0.00012) {
		tmp = fma((1.0 - cos(y)), fma((sqrt(2.0) * x), (1.00390625 * sin(y)), ((pow(sin(y), 2.0) * -0.0625) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0))
	tmp = 0.0
	if (x <= -0.000145)
		tmp = t_0;
	elseif (x <= 0.00012)
		tmp = Float64(fma(Float64(1.0 - cos(y)), fma(Float64(sqrt(2.0) * x), Float64(1.00390625 * sin(y)), Float64(Float64((sin(y) ^ 2.0) * -0.0625) * sqrt(2.0))), 2.0) / fma(fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0, 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.000145], t$95$0, If[LessEqual[x, 0.00012], N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision] * N[(1.00390625 * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * 3.0 + 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-4 or 1.20000000000000003e-4 < 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 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 99.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -1.45e-4 < x < 1.20000000000000003e-4

   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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

Alternative 17: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.0007:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.00058:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          1.5
          (fma
           (cos y)
           (/ 4.0 (+ (sqrt 5.0) 3.0))
           (* (- (sqrt 5.0) 1.0) (cos x)))
          3.0))
        (t_1
         (/
          (fma
           (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          t_0)))
   (if (<= x -0.0007)
     t_1
     (if (<= x 0.00058)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	double t_1 = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / t_0;
	double tmp;
	if (x <= -0.0007) {
		tmp = t_1;
	} else if (x <= 0.00058) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0)
	t_1 = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / t_0)
	tmp = 0.0
	if (x <= -0.0007)
		tmp = t_1;
	elseif (x <= 0.00058)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.0007], t$95$1, If[LessEqual[x, 0.00058], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999993e-4 or 5.8e-4 < 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 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 99.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -6.99999999999999993e-4 < x < 5.8e-4

   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 y 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. Step-by-step derivation

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sqrt.f64 99.4

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

   10. Applied rewrites 99.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0007:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00058:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{5} - 1\right) \cdot \cos x\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, t\_0\right), 3\right)}\\ \mathbf{if}\;x \leq -0.0007:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.00058:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- (sqrt 5.0) 1.0) (cos x)))
        (t_1
         (/
          (fma
           (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma (cos y) (/ 4.0 (+ (sqrt 5.0) 3.0)) t_0) 3.0))))
   (if (<= x -0.0007)
     t_1
     (if (<= x 0.00058)
       (/
        (+
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (* (pow (sin y) 2.0) -0.0625))
         2.0)
        (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 = (sqrt(5.0) - 1.0) * cos(x);
	double t_1 = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(y), (4.0 / (sqrt(5.0) + 3.0)), t_0), 3.0);
	double tmp;
	if (x <= -0.0007) {
		tmp = t_1;
	} else if (x <= 0.00058) {
		tmp = ((((1.0 - cos(y)) * sqrt(2.0)) * (pow(sin(y), 2.0) * -0.0625)) + 2.0) / 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(Float64(sqrt(5.0) - 1.0) * cos(x))
	t_1 = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), Float64(4.0 / Float64(sqrt(5.0) + 3.0)), t_0), 3.0))
	tmp = 0.0
	if (x <= -0.0007)
		tmp = t_1;
	elseif (x <= 0.00058)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * Float64((sin(y) ^ 2.0) * -0.0625)) + 2.0) / 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[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(N[Sqrt[5.0], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0007], t$95$1, If[LessEqual[x, 0.00058], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $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 < -6.99999999999999993e-4 or 5.8e-4 < 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 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 99.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 60.4

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

   10. Applied rewrites 60.4%

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

   if -6.99999999999999993e-4 < x < 5.8e-4

   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 y 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sqrt.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0007:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00058:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, \frac{4}{\sqrt{5} + 3}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{t\_0}\\ \mathbf{if}\;x \leq -0.0007:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 0.00058:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          1.5
          (fma (cos y) (- 3.0 (sqrt 5.0)) (* (- (sqrt 5.0) 1.0) (cos x)))
          3.0))
        (t_1
         (/
          (fma
           (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0))
           (pow (sin x) 2.0)
           2.0)
          t_0)))
   (if (<= x -0.0007)
     t_1
     (if (<= x 0.00058)
       (/
        (+
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (* (pow (sin y) 2.0) -0.0625))
         2.0)
        t_0)
       t_1))))

C

double code(double x, double y) {
	double t_0 = fma(1.5, fma(cos(y), (3.0 - sqrt(5.0)), ((sqrt(5.0) - 1.0) * cos(x))), 3.0);
	double t_1 = fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), pow(sin(x), 2.0), 2.0) / t_0;
	double tmp;
	if (x <= -0.0007) {
		tmp = t_1;
	} else if (x <= 0.00058) {
		tmp = ((((1.0 - cos(y)) * sqrt(2.0)) * (pow(sin(y), 2.0) * -0.0625)) + 2.0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(1.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), Float64(Float64(sqrt(5.0) - 1.0) * cos(x))), 3.0)
	t_1 = Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / t_0)
	tmp = 0.0
	if (x <= -0.0007)
		tmp = t_1;
	elseif (x <= 0.00058)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * Float64((sin(y) ^ 2.0) * -0.0625)) + 2.0) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -0.0007], t$95$1, If[LessEqual[x, 0.00058], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.99999999999999993e-4 or 5.8e-4 < 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 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. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 60.3

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

   8. Applied rewrites 60.3%

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

   if -6.99999999999999993e-4 < x < 5.8e-4

   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 y 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sqrt.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0007:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 0.00058:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0 \cdot \cos x\right), 3\right)}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))
           (pow (sin x) 2.0)
           2.0)
          (fma 1.5 (fma (cos y) t_1 (* t_0 (cos x))) 3.0))))
   (if (<= x -1.55e-6)
     t_2
     (if (<= x 7.8e-7)
       (/
        (+
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (* (pow (sin y) 2.0) -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)), pow(sin(x), 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, (t_0 * cos(x))), 3.0);
	double tmp;
	if (x <= -1.55e-6) {
		tmp = t_2;
	} else if (x <= 7.8e-7) {
		tmp = ((((1.0 - cos(y)) * sqrt(2.0)) * (pow(sin(y), 2.0) * -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(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), (sin(x) ^ 2.0), 2.0) / fma(1.5, fma(cos(y), t_1, Float64(t_0 * cos(x))), 3.0))
	tmp = 0.0
	if (x <= -1.55e-6)
		tmp = t_2;
	elseif (x <= 7.8e-7)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * Float64((sin(y) ^ 2.0) * -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[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1 + N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55e-6], t$95$2, If[LessEqual[x, 7.8e-7], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -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 < -1.55e-6 or 7.80000000000000049e-7 < 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 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. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-sin.f64 60.3

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

   8. Applied rewrites 60.3%

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

   if -1.55e-6 < x < 7.80000000000000049e-7

   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 y 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sqrt.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \left(\sqrt{5} - 1\right) \cdot \cos x\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \sqrt{5} - 1\\ t_2 := 3 - \sqrt{5}\\ t_3 := \mathsf{fma}\left(t\_1, \cos x, t\_2\right)\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, t\_0, 2\right)}{\mathsf{fma}\left(0.5, t\_3, 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_2, t\_1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0 \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, t\_3, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- (sqrt 5.0) 1.0))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (fma t_1 (cos x) t_2)))
   (if (<= x -7.5e-5)
     (*
      (/
       (fma (* (fma (cos x) -0.0625 0.0625) (sqrt 2.0)) t_0 2.0)
       (fma 0.5 t_3 1.0))
      0.3333333333333333)
     (if (<= x 1.65e-6)
       (/
        (+
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (* (pow (sin y) 2.0) -0.0625))
         2.0)
        (fma 1.5 (fma (cos y) t_2 t_1) 3.0))
       (/
        (fma (* t_0 (sqrt 2.0)) (fma -0.0625 (cos x) 0.0625) 2.0)
        (fma 1.5 t_3 3.0))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = sqrt(5.0) - 1.0;
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = fma(t_1, cos(x), t_2);
	double tmp;
	if (x <= -7.5e-5) {
		tmp = (fma((fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(0.5, t_3, 1.0)) * 0.3333333333333333;
	} else if (x <= 1.65e-6) {
		tmp = ((((1.0 - cos(y)) * sqrt(2.0)) * (pow(sin(y), 2.0) * -0.0625)) + 2.0) / fma(1.5, fma(cos(y), t_2, t_1), 3.0);
	} else {
		tmp = fma((t_0 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, t_3, 3.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(sqrt(5.0) - 1.0)
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = fma(t_1, cos(x), t_2)
	tmp = 0.0
	if (x <= -7.5e-5)
		tmp = Float64(Float64(fma(Float64(fma(cos(x), -0.0625, 0.0625) * sqrt(2.0)), t_0, 2.0) / fma(0.5, t_3, 1.0)) * 0.3333333333333333);
	elseif (x <= 1.65e-6)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * Float64((sin(y) ^ 2.0) * -0.0625)) + 2.0) / fma(1.5, fma(cos(y), t_2, t_1), 3.0));
	else
		tmp = Float64(fma(Float64(t_0 * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, t_3, 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Cos[x], $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[x, -7.5e-5], N[(N[(N[(N[(N[(N[Cos[x], $MachinePrecision] * -0.0625 + 0.0625), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] / N[(0.5 * t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], If[LessEqual[x, 1.65e-6], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2 + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * t$95$3 + 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999934e-5

   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 y 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. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 64.7%

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

   if -7.49999999999999934e-5 < x < 1.65000000000000008e-6

   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 y 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sqrt.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.4%

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

   if 1.65000000000000008e-6 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-cos.f64 54.2

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

   5. Applied rewrites 54.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 53.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos x, -0.0625, 0.0625\right) \cdot \sqrt{2}, {\sin x}^{2}, 2\right)}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 1\right)} \cdot 0.3333333333333333\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (fma 1.5 (fma t_0 (cos x) t_1) 3.0))))
   (if (<= x -7.5e-5)
     t_2
     (if (<= x 1.65e-6)
       (/
        (+
         (* (* (- 1.0 (cos y)) (sqrt 2.0)) (* (pow (sin y) 2.0) -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((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	double tmp;
	if (x <= -7.5e-5) {
		tmp = t_2;
	} else if (x <= 1.65e-6) {
		tmp = ((((1.0 - cos(y)) * sqrt(2.0)) * (pow(sin(y), 2.0) * -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(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0))
	tmp = 0.0
	if (x <= -7.5e-5)
		tmp = t_2;
	elseif (x <= 1.65e-6)
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 - cos(y)) * sqrt(2.0)) * Float64((sin(y) ^ 2.0) * -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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e-5], t$95$2, If[LessEqual[x, 1.65e-6], N[(N[(N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -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 < -7.49999999999999934e-5 or 1.65000000000000008e-6 < 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-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-cos.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 59.3%

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

   if -7.49999999999999934e-5 < x < 1.65000000000000008e-6

   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 y 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

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sqrt.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\left(1 - \cos y\right) \cdot \sqrt{2}\right) \cdot \left({\sin y}^{2} \cdot -0.0625\right) + 2}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} - 1\\ t_1 := 3 - \sqrt{5}\\ t_2 := \frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(t\_0, \cos x, t\_1\right), 3\right)}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, t\_1, t\_0\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt 5.0) 1.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2
         (/
          (fma
           (* (pow (sin x) 2.0) (sqrt 2.0))
           (fma -0.0625 (cos x) 0.0625)
           2.0)
          (fma 1.5 (fma t_0 (cos x) t_1) 3.0))))
   (if (<= x -7.5e-5)
     t_2
     (if (<= x 1.65e-6)
       (/
        (fma (* (pow (sin y) 2.0) -0.0625) (* (- 1.0 (cos y)) (sqrt 2.0)) 2.0)
        (fma 1.5 (fma (cos y) t_1 t_0) 3.0))
       t_2))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) - 1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = fma((pow(sin(x), 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0);
	double tmp;
	if (x <= -7.5e-5) {
		tmp = t_2;
	} else if (x <= 1.65e-6) {
		tmp = fma((pow(sin(y), 2.0) * -0.0625), ((1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(y), t_1, t_0), 3.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) - 1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(fma(Float64((sin(x) ^ 2.0) * sqrt(2.0)), fma(-0.0625, cos(x), 0.0625), 2.0) / fma(1.5, fma(t_0, cos(x), t_1), 3.0))
	tmp = 0.0
	if (x <= -7.5e-5)
		tmp = t_2;
	elseif (x <= 1.65e-6)
		tmp = Float64(fma(Float64((sin(y) ^ 2.0) * -0.0625), Float64(Float64(1.0 - cos(y)) * sqrt(2.0)), 2.0) / fma(1.5, fma(cos(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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[Cos[x], $MachinePrecision] + 0.0625), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(t$95$0 * N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e-5], t$95$2, If[LessEqual[x, 1.65e-6], N[(N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(1.5 * N[(N[Cos[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 < -7.49999999999999934e-5 or 1.65000000000000008e-6 < 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-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. sub-neg N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-cos.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 59.3%

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

   if -7.49999999999999934e-5 < x < 1.65000000000000008e-6

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+l+ N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

   7. Applied rewrites 99.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin y}^{2} \cdot -0.0625, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5} - 1\right), 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\sin x}^{2} \cdot \sqrt{2}, \mathsf{fma}\left(-0.0625, \cos x, 0.0625\right), 2\right)}{\mathsf{fma}\left(1.5, \mathsf{fma}\left(\sqrt{5} - 1, \cos x, 3 - \sqrt{5}\right), 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 60.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
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-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. sub-neg N/A

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

   13. distribute-lft-in N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-cos.f64 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. distribute-lft-out N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

8. Applied rewrites 61.0%

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

9. Final simplification 61.0%

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

Alternative 25: 45.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sqrt.f64 63.1

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

7. Applied rewrites 63.1%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 46.5%

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

12. Applied rewrites 46.5%

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

Alternative 26: 45.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. distribute-rgt-in N/A

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

   6. metadata-eval N/A

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

   7. lower-+.f64 N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   11. lower-sqrt.f64 63.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

7. Applied rewrites 63.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
9. Step-by-step derivation

10. Applied rewrites 46.5%

    \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

12. Applied rewrites 46.5%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\left(3 - \sqrt{5}\right) \cdot 0.5, \cos y \cdot 3, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)}} \]

13. Final simplification 46.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left(0.5 \cdot \left(3 - \sqrt{5}\right), 3 \cdot \cos y, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right) \cdot 3\right)} \]
14. Add Preprocessing

Alternative 27: 45.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right) \cdot 3 + 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (+
   (*
    (fma
     (cos y)
     (* 0.5 (- 3.0 (sqrt 5.0)))
     (* (fma (sqrt 5.0) 0.5 -0.5) (cos x)))
    3.0)
   3.0)))

C

double code(double x, double y) {
	return 2.0 / ((fma(cos(y), (0.5 * (3.0 - sqrt(5.0))), (fma(sqrt(5.0), 0.5, -0.5) * cos(x))) * 3.0) + 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(Float64(fma(cos(y), Float64(0.5 * Float64(3.0 - sqrt(5.0))), Float64(fma(sqrt(5.0), 0.5, -0.5) * cos(x))) * 3.0) + 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[Cos[y], $MachinePrecision] * N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   11. lower-sqrt.f64 63.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

7. Applied rewrites 63.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
9. Step-by-step derivation

10. Applied rewrites 46.5%

    \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

11. Final simplification 46.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right) \cdot \cos x\right) \cdot 3 + 3} \]
12. Add Preprocessing

Alternative 28: 45.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (*
   (fma
    (* 0.5 (cos y))
    (- 3.0 (sqrt 5.0))
    (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) 1.0))
   3.0)))

C

double code(double x, double y) {
	return 2.0 / (fma((0.5 * cos(y)), (3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(fma(Float64(0.5 * cos(y)), Float64(3.0 - sqrt(5.0)), fma(fma(0.5, sqrt(5.0), -0.5), cos(x), 1.0)) * 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(0.5 * N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + 1.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   11. lower-sqrt.f64 63.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

7. Applied rewrites 63.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
9. Step-by-step derivation

10. Applied rewrites 46.5%

    \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

12. Applied rewrites 46.5%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(0.5 \cdot \cos y, 3 - \sqrt{5}, \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 1\right)\right) \cdot 3}} \]
13. Add Preprocessing

Alternative 29: 42.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3 + 3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (+
   (* (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (* 0.5 (- 3.0 (sqrt 5.0)))) 3.0)
   3.0)))

C

double code(double x, double y) {
	return 2.0 / ((fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (0.5 * (3.0 - sqrt(5.0)))) * 3.0) + 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / Float64(Float64(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(0.5 * Float64(3.0 - sqrt(5.0)))) * 3.0) + 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   11. lower-sqrt.f64 63.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

7. Applied rewrites 63.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
9. Step-by-step derivation

10. Applied rewrites 46.5%

    \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

11. Taylor expanded in y around 0

    \[\leadsto \frac{2}{3 + \color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)} \cdot 3} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{3 + \color{blue}{\left(\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot 3} \]

    2. *-commutative N/A

       \[\leadsto \frac{2}{3 + \left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot \cos x} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{2}{3 + \color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)} \cdot 3} \]

    4. sub-neg N/A

       \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3} \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3} \]

    7. lower-sqrt.f64 N/A

       \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{5}}, \frac{-1}{2}\right), \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3} \]

    8. lower-cos.f64 N/A

       \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \color{blue}{\cos x}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3} \]

    9. *-commutative N/A

       \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}}\right) \cdot 3} \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}}\right) \cdot 3} \]

    11. lower--.f64 N/A

        \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right) \cdot 3} \]

    12. lower-sqrt.f64 44.3

        \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot 0.5\right) \cdot 3} \]

13. Applied rewrites 44.3%

    \[\leadsto \frac{2}{3 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot 0.5\right)} \cdot 3} \]

14. Final simplification 44.3%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 0.5 \cdot \left(3 - \sqrt{5}\right)\right) \cdot 3 + 3} \]
15. Add Preprocessing

Alternative 30: 42.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma
   (fma (fma 0.5 (sqrt 5.0) -0.5) (cos x) (* 0.5 (- 3.0 (sqrt 5.0))))
   3.0
   3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), (0.5 * (3.0 - sqrt(5.0)))), 3.0, 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(fma(fma(0.5, sqrt(5.0), -0.5), cos(x), Float64(0.5 * Float64(3.0 - sqrt(5.0)))), 3.0, 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(0.5 * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0 + 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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   11. lower-sqrt.f64 63.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

7. Applied rewrites 63.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
9. Step-by-step derivation

10. Applied rewrites 46.5%

    \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

11. Taylor expanded in y around 0

    \[\leadsto \frac{2}{\color{blue}{3 + 3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right)}} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) + 3}} \]

    2. *-commutative N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right)\right) \cdot 3} + 3} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(3 - \sqrt{5}\right) + \cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right), 3, 3\right)}} \]

    4. +-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\cos x \cdot \left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)}, 3, 3\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}\right) \cdot \cos x} + \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), 3, 3\right)} \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} - \frac{1}{2}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right)}, 3, 3\right)} \]

    7. sub-neg N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sqrt{5} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{-1}{2}}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)} \]

    9. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right)}, \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)} \]

    10. lower-sqrt.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sqrt{5}}, \frac{-1}{2}\right), \cos x, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)} \]

    11. lower-cos.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \color{blue}{\cos x}, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)} \]

    12. *-commutative N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}}\right), 3, 3\right)} \]

    13. lower-*.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right) \cdot \frac{1}{2}}\right), 3, 3\right)} \]

    14. lower--.f64 N/A

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \sqrt{5}, \frac{-1}{2}\right), \cos x, \color{blue}{\left(3 - \sqrt{5}\right)} \cdot \frac{1}{2}\right), 3, 3\right)} \]

    15. lower-sqrt.f64 44.3

        \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(3 - \color{blue}{\sqrt{5}}\right) \cdot 0.5\right), 3, 3\right)} \]

13. Applied rewrites 44.3%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, \left(3 - \sqrt{5}\right) \cdot 0.5\right), 3, 3\right)}} \]

14. Final simplification 44.3%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \cos x, 0.5 \cdot \left(3 - \sqrt{5}\right)\right), 3, 3\right)} \]
15. Add Preprocessing

Alternative 31: 42.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3, 3\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  2.0
  (fma (fma 0.5 (fma (cos y) (- 3.0 (sqrt 5.0)) (sqrt 5.0)) -0.5) 3.0 3.0)))

C

double code(double x, double y) {
	return 2.0 / fma(fma(0.5, fma(cos(y), (3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0, 3.0);
}

Julia

function code(x, y)
	return Float64(2.0 / fma(fma(0.5, fma(cos(y), Float64(3.0 - sqrt(5.0)), sqrt(5.0)), -0.5), 3.0, 3.0))
end

Wolfram

code[x_, y_] := N[(2.0 / N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * 3.0 + 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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\color{blue}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right)} + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   4. associate-+l+ N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \color{blue}{\left(1 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)\right)}} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{1 \cdot 3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3} + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 + \color{blue}{\left(\frac{\sqrt{5} - 1}{2} \cdot \cos x + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right) \cdot 3}} \]

4. Applied rewrites 99.4%

   \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{\color{blue}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2 + \frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right) + 2}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{16} \cdot {\sin y}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)} + 2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{16} \cdot {\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot \color{blue}{{\sin y}^{2}}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   6. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\color{blue}{\sin y}}^{2}, \sqrt{2} \cdot \left(1 - \cos y\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right) \cdot \sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \color{blue}{\left(1 - \cos y\right)} \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   10. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{16} \cdot {\sin y}^{2}, \left(1 - \color{blue}{\cos y}\right) \cdot \sqrt{2}, 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]

   11. lower-sqrt.f64 63.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \color{blue}{\sqrt{2}}, 2\right)}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

7. Applied rewrites 63.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0625 \cdot {\sin y}^{2}, \left(1 - \cos y\right) \cdot \sqrt{2}, 2\right)}}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, \frac{1}{2} \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, \frac{1}{2}, \frac{-1}{2}\right)\right) \cdot 3} \]
9. Step-by-step derivation

10. Applied rewrites 46.5%

    \[\leadsto \frac{2}{3 + \mathsf{fma}\left(\cos y, 0.5 \cdot \left(3 - \sqrt{5}\right), \cos x \cdot \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)\right) \cdot 3} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{2}{\color{blue}{3 + 3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right)}} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{2}{\color{blue}{3 \cdot \left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right) + 3}} \]

    2. *-commutative N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) - \frac{1}{2}\right) \cdot 3} + 3} \]

    3. sub-neg N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{1}{2} \cdot \sqrt{5} + \frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot 3 + 3} \]

    4. +-commutative N/A

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \sqrt{5}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot 3 + 3} \]

    5. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \sqrt{5}\right) + \color{blue}{\frac{-1}{2}}\right) \cdot 3 + 3} \]

    6. associate-+l+ N/A

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(\frac{1}{2} \cdot \sqrt{5} + \frac{-1}{2}\right)\right)} \cdot 3 + 3} \]

    7. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(\frac{1}{2} \cdot \sqrt{5} + \color{blue}{\frac{1}{2} \cdot -1}\right)\right) \cdot 3 + 3} \]

    8. distribute-lft-in N/A

       \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{5} + -1\right)}\right) \cdot 3 + 3} \]

    9. metadata-eval N/A

       \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \cdot 3 + 3} \]

    10. sub-neg N/A

        \[\leadsto \frac{2}{\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \color{blue}{\left(\sqrt{5} - 1\right)}\right) \cdot 3 + 3} \]

    11. lower-fma.f64 N/A

        \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \frac{1}{2} \cdot \left(\sqrt{5} - 1\right), 3, 3\right)}} \]

13. Applied rewrites 43.5%

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos y, 3 - \sqrt{5}, \sqrt{5}\right), -0.5\right), 3, 3\right)}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(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))))))