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

Percentage Accurate: 99.3% → 99.4%

Time: 47.1s

Alternatives: 30

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, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin y) (* -0.0625 (sin x)))
    (* (- (cos x) (cos y)) (+ (sin x) (* (sin y) -0.0625))))
   2.0)
  (+
   3.0
   (fma
    1.5
    (* (cos x) (+ (sqrt 5.0) -1.0))
    (/ (* (cos y) 6.0) (+ 3.0 (sqrt 5.0)))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) + (-0.0625 * sin(x))) * ((cos(x) - cos(y)) * (sin(x) + (sin(y) * -0.0625)))), 2.0) / (3.0 + fma(1.5, (cos(x) * (sqrt(5.0) + -1.0)), ((cos(y) * 6.0) / (3.0 + sqrt(5.0)))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(-0.0625 * sin(x))) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) + Float64(sin(y) * -0.0625)))), 2.0) / Float64(3.0 + fma(1.5, Float64(cos(x) * Float64(sqrt(5.0) + -1.0)), Float64(Float64(cos(y) * 6.0) / Float64(3.0 + sqrt(5.0))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[y], $MachinePrecision] * 6.0), $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
4. Step-by-step derivation

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   3. add-sqr-sqrt 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   4. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

7. Simplified 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

8. Taylor expanded in y around inf 99.3%

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}} \]

10. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{\cos y \cdot 6}{3 + \sqrt{5}}\right)} \]

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \left(\frac{6}{\frac{3 + \sqrt{5}}{\cos y}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) + (-0.0625 * sin(x))) * ((cos(x) - cos(y)) * (sin(x) + (sin(y) * -0.0625)))), 2.0) / (3.0 + ((6.0 / ((3.0 + sqrt(5.0)) / cos(y))) + (1.5 * (cos(x) * (sqrt(5.0) + -1.0)))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(-0.0625 * sin(x))) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) + Float64(sin(y) * -0.0625)))), 2.0) / Float64(3.0 + Float64(Float64(6.0 / Float64(Float64(3.0 + sqrt(5.0)) / cos(y))) + Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(6.0 / N[(N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
4. Step-by-step derivation

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   3. add-sqr-sqrt 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   4. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

7. Simplified 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

8. Taylor expanded in y around inf 99.3%

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}} \]

10. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right)\right), 2\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}} \]
11. Step-by-step derivation

    1. fma-udef 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right)\right), 2\right)}{3 + \color{blue}{\left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \frac{6 \cdot \cos y}{3 + \sqrt{5}}\right)}} \]

    2. associate-/l* 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \color{blue}{\frac{6}{\frac{3 + \sqrt{5}}{\cos y}}}\right)} \]

    3. +-commutative 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \frac{6}{\frac{\color{blue}{\sqrt{5} + 3}}{\cos y}}\right)} \]

12. Applied egg-rr 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + -0.0625 \cdot \sin y\right)\right), 2\right)}{3 + \color{blue}{\left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + \frac{6}{\frac{\sqrt{5} + 3}{\cos y}}\right)}} \]

13. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x + \sin y \cdot -0.0625\right)\right), 2\right)}{3 + \left(\frac{6}{\frac{3 + \sqrt{5}}{\cos y}} + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \]

Alternative 3: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := \cos x - \cos y\\ t_4 := \sqrt{2} \cdot \sin x\\ \mathbf{if}\;x \leq -0.065:\\ \;\;\;\;\frac{2 + t_3 \cdot \left(t_4 \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.085:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_2 - 0.5\right) + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(t_3 \cdot t_0\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + t_1, \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (/ (sqrt 5.0) 2.0))
        (t_3 (- (cos x) (cos y)))
        (t_4 (* (sqrt 2.0) (sin x))))
   (if (<= x -0.065)
     (/
      (+ 2.0 (* t_3 (* t_4 t_0)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 0.085)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (* t_0 (+ 1.0 (- (* -0.5 (* x x)) (cos y))))))
        (* 3.0 (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))
       (/
        (+ 2.0 (* t_4 (* t_3 t_0)))
        (* 3.0 (+ 1.0 (fma (cos x) (+ -0.5 t_1) (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = sqrt(5.0) / 2.0;
	double t_3 = cos(x) - cos(y);
	double t_4 = sqrt(2.0) * sin(x);
	double tmp;
	if (x <= -0.065) {
		tmp = (2.0 + (t_3 * (t_4 * t_0))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 0.085) {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - cos(y)))))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	} else {
		tmp = (2.0 + (t_4 * (t_3 * t_0))) / (3.0 * (1.0 + fma(cos(x), (-0.5 + t_1), (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(sqrt(5.0) / 2.0)
	t_3 = Float64(cos(x) - cos(y))
	t_4 = Float64(sqrt(2.0) * sin(x))
	tmp = 0.0
	if (x <= -0.065)
		tmp = Float64(Float64(2.0 + Float64(t_3 * Float64(t_4 * t_0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (x <= 0.085)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_0 * Float64(1.0 + Float64(Float64(-0.5 * Float64(x * x)) - cos(y)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_4 * Float64(t_3 * t_0))) / Float64(3.0 * Float64(1.0 + fma(cos(x), Float64(-0.5 + t_1), Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.065], N[(N[(2.0 + N[(t$95$3 * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.085], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$4 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.065000000000000002

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 63.8%

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

      1. *-commutative 63.8%

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

   4. Simplified 63.8%

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

   if -0.065000000000000002 < x < 0.0850000000000000061

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.4%

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

      2. unpow2 99.4%

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

   6. Simplified 99.4%

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

   if 0.0850000000000000061 < 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. Step-by-step derivation

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

      1. fma-def 99.0%

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

      2. sub-neg 99.0%

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

      3. div-inv 99.0%

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

      4. metadata-eval 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 62.7%

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

      1. *-commutative 62.7%

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

   8. Simplified 62.7%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.065:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.085:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * ((sin(y) + (-0.0625 * sin(x))) * (cos(x) - cos(y)))))) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
}

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) * (-0.0625d0))) * ((sin(y) + ((-0.0625d0) * sin(x))) * (cos(x) - cos(y)))))) / (3.0d0 + ((1.5d0 * (cos(x) * (sqrt(5.0d0) + (-1.0d0)))) + (6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0))))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (Math.sin(y) * -0.0625)) * ((Math.sin(y) + (-0.0625 * Math.sin(x))) * (Math.cos(x) - Math.cos(y)))))) / (3.0 + ((1.5 * (Math.cos(x) * (Math.sqrt(5.0) + -1.0))) + (6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0))))));
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (math.sin(y) * -0.0625)) * ((math.sin(y) + (-0.0625 * math.sin(x))) * (math.cos(x) - math.cos(y)))))) / (3.0 + ((1.5 * (math.cos(x) * (math.sqrt(5.0) + -1.0))) + (6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0))))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(Float64(sin(y) + Float64(-0.0625 * sin(x))) * Float64(cos(x) - cos(y)))))) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * Float64(sqrt(5.0) + -1.0))) + Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * ((sin(y) + (-0.0625 * sin(x))) * (cos(x) - cos(y)))))) / (3.0 + ((1.5 * (cos(x) * (sqrt(5.0) + -1.0))) + (6.0 * (cos(y) / (3.0 + sqrt(5.0))))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(-0.0625 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
4. Step-by-step derivation

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   3. add-sqr-sqrt 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   4. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
6. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

7. Simplified 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

8. Taylor expanded in y around inf 99.3%

   \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}} \]

9. Final simplification 99.3%

   \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \]

Alternative 5: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.112 \lor \neg \left(x \leq 0.082\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0))) (t_1 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.112) (not (<= x 0.082)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* (* (sqrt 2.0) (sin x)) t_0)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
        (* t_0 (+ 1.0 (- (* -0.5 (* x x)) (cos y))))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.112) || !(x <= 0.082)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - cos(y)))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = sqrt(5.0d0) / 2.0d0
    if ((x <= (-0.112d0)) .or. (.not. (x <= 0.082d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * t_0))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (t_0 * (1.0d0 + (((-0.5d0) * (x * x)) - cos(y)))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_1 - 0.5d0)) + (cos(y) * (1.5d0 - t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.112) || !(x <= 0.082)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * t_0))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - Math.cos(y)))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_1 - 0.5)) + (Math.cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.112) or not (x <= 0.082):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * t_0))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - math.cos(y)))))) / (3.0 * (1.0 + ((math.cos(x) * (t_1 - 0.5)) + (math.cos(y) * (1.5 - t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.112) || !(x <= 0.082))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * t_0))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_0 * Float64(1.0 + Float64(Float64(-0.5 * Float64(x * x)) - cos(y)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.112) || ~((x <= 0.082)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - cos(y)))))) / (3.0 * (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.112], N[Not[LessEqual[x, 0.082]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.112000000000000002 or 0.0820000000000000034 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   4. Simplified 63.3%

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

   if -0.112000000000000002 < x < 0.0820000000000000034

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. associate--l+ 99.4%

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

      2. unpow2 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.112 \lor \neg \left(x \leq 0.082\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.056\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= x -0.029) (not (<= x 0.056)))
     (/ (+ 2.0 (* t_2 (* (* (sqrt 2.0) (sin x)) t_0))) t_1)
     (/
      (+ 2.0 (* t_2 (* t_0 (* (sqrt 2.0) (+ x (* (sin y) -0.0625))))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_2 = cos(x) - cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.056)) {
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    t_2 = cos(x) - cos(y)
    if ((x <= (-0.029d0)) .or. (.not. (x <= 0.056d0))) then
        tmp = (2.0d0 + (t_2 * ((sqrt(2.0d0) * sin(x)) * t_0))) / t_1
    else
        tmp = (2.0d0 + (t_2 * (t_0 * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double t_2 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.056)) {
		tmp = (2.0 + (t_2 * ((Math.sqrt(2.0) * Math.sin(x)) * t_0))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	t_2 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -0.029) or not (x <= 0.056):
		tmp = (2.0 + (t_2 * ((math.sqrt(2.0) * math.sin(x)) * t_0))) / t_1
	else:
		tmp = (2.0 + (t_2 * (t_0 * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -0.029) || !(x <= 0.056))
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(sqrt(2.0) * sin(x)) * t_0))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	t_2 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -0.029) || ~((x <= 0.056)))
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_0))) / t_1;
	else
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.029], N[Not[LessEqual[x, 0.056]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 0.0560000000000000012 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   4. Simplified 63.3%

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

   if -0.0290000000000000015 < x < 0.0560000000000000012

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right) + x \cdot \sqrt{2}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 99.1%

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

      2. *-commutative 99.1%

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

      3. distribute-rgt-out 99.1%

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

      4. *-commutative 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.056\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.0026\right):\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(x \cdot \left(\sin y \cdot 1.00390625\right) + -0.0625 \cdot {\sin y}^{2}\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_1 (- (cos x) (cos y))))
   (if (or (<= x -0.029) (not (<= x 0.0026)))
     (/
      (+ 2.0 (* t_1 (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      t_0)
     (/
      (+
       2.0
       (*
        t_1
        (*
         (sqrt 2.0)
         (+ (* x (* (sin y) 1.00390625)) (* -0.0625 (pow (sin y) 2.0))))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_1 = cos(x) - cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.0026)) {
		tmp = (2.0 + (t_1 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / t_0;
	} else {
		tmp = (2.0 + (t_1 * (sqrt(2.0) * ((x * (sin(y) * 1.00390625)) + (-0.0625 * pow(sin(y), 2.0)))))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    t_1 = cos(x) - cos(y)
    if ((x <= (-0.029d0)) .or. (.not. (x <= 0.0026d0))) then
        tmp = (2.0d0 + (t_1 * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / t_0
    else
        tmp = (2.0d0 + (t_1 * (sqrt(2.0d0) * ((x * (sin(y) * 1.00390625d0)) + ((-0.0625d0) * (sin(y) ** 2.0d0)))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double t_1 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.0026)) {
		tmp = (2.0 + (t_1 * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / t_0;
	} else {
		tmp = (2.0 + (t_1 * (Math.sqrt(2.0) * ((x * (Math.sin(y) * 1.00390625)) + (-0.0625 * Math.pow(Math.sin(y), 2.0)))))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	t_1 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -0.029) or not (x <= 0.0026):
		tmp = (2.0 + (t_1 * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / t_0
	else:
		tmp = (2.0 + (t_1 * (math.sqrt(2.0) * ((x * (math.sin(y) * 1.00390625)) + (-0.0625 * math.pow(math.sin(y), 2.0)))))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_1 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -0.029) || !(x <= 0.0026))
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(sqrt(2.0) * Float64(Float64(x * Float64(sin(y) * 1.00390625)) + Float64(-0.0625 * (sin(y) ^ 2.0)))))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	t_1 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -0.029) || ~((x <= 0.0026)))
		tmp = (2.0 + (t_1 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / t_0;
	else
		tmp = (2.0 + (t_1 * (sqrt(2.0) * ((x * (sin(y) * 1.00390625)) + (-0.0625 * (sin(y) ^ 2.0)))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.029], N[Not[LessEqual[x, 0.0026]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$1 * 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]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(x * N[(N[Sin[y], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 0.0025999999999999999 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   4. Simplified 63.3%

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

   if -0.0290000000000000015 < x < 0.0025999999999999999

   1. Initial program 99.6%

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

      1. add-log-exp 99.7%

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

      2. div-inv 99.7%

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

      3. metadata-eval 99.7%

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

      4. div-inv 99.7%

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

      5. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 98.7%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right) + x \cdot \left(\sqrt{2} \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   5. Step-by-step derivation

      1. +-commutative 98.7%

         \[\leadsto \frac{2 + \color{blue}{\left(x \cdot \left(\sqrt{2} \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right) + -0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      2. *-commutative 98.7%

         \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{2} \cdot \left(\sin y + 0.00390625 \cdot \sin y\right)\right) \cdot x} + -0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      3. associate-*l* 98.7%

         \[\leadsto \frac{2 + \left(\color{blue}{\sqrt{2} \cdot \left(\left(\sin y + 0.00390625 \cdot \sin y\right) \cdot x\right)} + -0.0625 \cdot \left({\sin y}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      4. associate-*r* 98.7%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y + 0.00390625 \cdot \sin y\right) \cdot x\right) + \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \sqrt{2}}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      5. *-commutative 98.7%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\left(\sin y + 0.00390625 \cdot \sin y\right) \cdot x\right) + \color{blue}{\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      6. distribute-lft-out 98.7%

         \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot \left(\left(\sin y + 0.00390625 \cdot \sin y\right) \cdot x + -0.0625 \cdot {\sin y}^{2}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      7. distribute-rgt1-in 98.7%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\color{blue}{\left(\left(0.00390625 + 1\right) \cdot \sin y\right)} \cdot x + -0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      8. *-commutative 98.7%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\color{blue}{\left(\sin y \cdot \left(0.00390625 + 1\right)\right)} \cdot x + -0.0625 \cdot {\sin y}^{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

      9. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.0026\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(x \cdot \left(\sin y \cdot 1.00390625\right) + -0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 8: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00022\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot t_1\right)}{3 \cdot \left(1 + \left(\left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= x -0.029) (not (<= x 0.00022)))
     (/
      (+ 2.0 (* t_2 (* (* (sqrt 2.0) (sin x)) t_1)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+ 2.0 (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (* t_2 t_1)))
      (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = cos(x) - cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00022)) {
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_1))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = sin(y) - (sin(x) / 16.0d0)
    t_2 = cos(x) - cos(y)
    if ((x <= (-0.029d0)) .or. (.not. (x <= 0.00022d0))) then
        tmp = (2.0d0 + (t_2 * ((sqrt(2.0d0) * sin(x)) * t_1))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (t_2 * t_1))) / (3.0d0 * (1.0d0 + ((t_0 + (cos(y) * (1.5d0 - t_0))) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_2 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00022)) {
		tmp = (2.0 + (t_2 * ((Math.sqrt(2.0) * Math.sin(x)) * t_1))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (Math.cos(y) * (1.5 - t_0))) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.sin(y) - (math.sin(x) / 16.0)
	t_2 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -0.029) or not (x <= 0.00022):
		tmp = (2.0 + (t_2 * ((math.sqrt(2.0) * math.sin(x)) * t_1))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (math.cos(y) * (1.5 - t_0))) - 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -0.029) || !(x <= 0.00022))
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(sqrt(2.0) * sin(x)) * t_1))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_2 * t_1))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = sin(y) - (sin(x) / 16.0);
	t_2 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -0.029) || ~((x <= 0.00022)))
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_1))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.029], N[Not[LessEqual[x, 0.00022]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 2.20000000000000008e-4 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   4. Simplified 63.3%

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

   if -0.0290000000000000015 < x < 2.20000000000000008e-4

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 98.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00022\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 9: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00094\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_2 \cdot t_1\right)}{3 \cdot \left(1 + \left(\left(t_0 + \frac{\cos y}{1.5 + t_0}\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= x -0.029) (not (<= x 0.00094)))
     (/
      (+ 2.0 (* t_2 (* (* (sqrt 2.0) (sin x)) t_1)))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (+ 2.0 (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (* t_2 t_1)))
      (* 3.0 (+ 1.0 (- (+ t_0 (/ (cos y) (+ 1.5 t_0))) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = cos(x) - cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00094)) {
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_1))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (cos(y) / (1.5 + t_0))) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = sin(y) - (sin(x) / 16.0d0)
    t_2 = cos(x) - cos(y)
    if ((x <= (-0.029d0)) .or. (.not. (x <= 0.00094d0))) then
        tmp = (2.0d0 + (t_2 * ((sqrt(2.0d0) * sin(x)) * t_1))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (t_2 * t_1))) / (3.0d0 * (1.0d0 + ((t_0 + (cos(y) / (1.5d0 + t_0))) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_2 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00094)) {
		tmp = (2.0 + (t_2 * ((Math.sqrt(2.0) * Math.sin(x)) * t_1))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (Math.cos(y) / (1.5 + t_0))) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.sin(y) - (math.sin(x) / 16.0)
	t_2 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -0.029) or not (x <= 0.00094):
		tmp = (2.0 + (t_2 * ((math.sqrt(2.0) * math.sin(x)) * t_1))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (math.cos(y) / (1.5 + t_0))) - 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -0.029) || !(x <= 0.00094))
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(sqrt(2.0) * sin(x)) * t_1))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_2 * t_1))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) / Float64(1.5 + t_0))) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = sin(y) - (sin(x) / 16.0);
	t_2 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -0.029) || ~((x <= 0.00094)))
		tmp = (2.0 + (t_2 * ((sqrt(2.0) * sin(x)) * t_1))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_2 * t_1))) / (3.0 * (1.0 + ((t_0 + (cos(y) / (1.5 + t_0))) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.029], N[Not[LessEqual[x, 0.00094]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$2 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] / N[(1.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 9.39999999999999972e-4 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   4. Simplified 63.3%

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

   if -0.0290000000000000015 < x < 9.39999999999999972e-4

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00094\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \frac{\cos y}{1.5 + \sqrt{5} \cdot 0.5}\right) - 0.5\right)\right)}\\ \end{array} \]

Alternative 10: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 0.00105\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0009) (not (<= x 0.00105)))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (sin x))
        (* (- (cos x) (cos y)) (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (fma (sqrt 2.0) (* (* -0.0625 (pow (sin y) 2.0)) (- 1.0 (cos y))) 2.0)
      (+
       (/ (* (cos x) (+ (sqrt 5.0) -1.0)) 0.6666666666666666)
       (fma (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 0.6666666666666666) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0009) || !(x <= 0.00105)) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((cos(x) - cos(y)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = fma(sqrt(2.0), ((-0.0625 * pow(sin(y), 2.0)) * (1.0 - cos(y))), 2.0) / (((cos(x) * (sqrt(5.0) + -1.0)) / 0.6666666666666666) + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) / 0.6666666666666666), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0009) || !(x <= 0.00105))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(1.0 - cos(y))), 2.0) / Float64(Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) / 0.6666666666666666) + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 0.6666666666666666), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0009], N[Not[LessEqual[x, 0.00105]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999998e-4 or 0.00104999999999999994 < x

   1. Initial program 98.8%

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

      1. associate-*l* 98.8%

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

      2. distribute-lft-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-lft-in 98.8%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 63.1%

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

      1. *-commutative 63.1%

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

   6. Simplified 63.1%

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

   if -8.9999999999999998e-4 < x < 0.00104999999999999994

   1. Initial program 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

      2. *-commutative 98.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

   10. Simplified 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 0.00105\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)}\\ \end{array} \]

Alternative 11: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 0.00086\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\frac{\cos x \cdot t_0}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.0009) (not (<= x 0.00086)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (/
      (fma (sqrt 2.0) (* (* -0.0625 (pow (sin y) 2.0)) (- 1.0 (cos y))) 2.0)
      (+
       (/ (* (cos x) t_0) 0.6666666666666666)
       (fma (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 0.6666666666666666) 3.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.0009) || !(x <= 0.00086)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), ((-0.0625 * pow(sin(y), 2.0)) * (1.0 - cos(y))), 2.0) / (((cos(x) * t_0) / 0.6666666666666666) + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) / 0.6666666666666666), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.0009) || !(x <= 0.00086))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(1.0 - cos(y))), 2.0) / Float64(Float64(Float64(cos(x) * t_0) / 0.6666666666666666) + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 0.6666666666666666), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.0009], N[Not[LessEqual[x, 0.00086]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999998e-4 or 8.59999999999999979e-4 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 63.1%

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

      1. *-commutative 63.1%

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

   4. Simplified 63.1%

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

   if -8.9999999999999998e-4 < x < 8.59999999999999979e-4

   1. Initial program 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

      2. *-commutative 98.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

   10. Simplified 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 0.00086\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)}\\ \end{array} \]

Alternative 12: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{\cos x \cdot t_0}{0.6666666666666666}\\ t_2 := \cos x + -1\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0012:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t_2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_3}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00072:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{t_1 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_2 \cdot \left(-0.0625 \cdot {\sin x}^{2}\right), 2\right)}{t_1 + \mathsf{fma}\left(\cos y, \frac{t_3}{0.6666666666666666}, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (/ (* (cos x) t_0) 0.6666666666666666))
        (t_2 (+ (cos x) -1.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0012)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))) t_2))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_3 2.0)))))
     (if (<= x 0.00072)
       (/
        (fma (sqrt 2.0) (* (* -0.0625 (pow (sin y) 2.0)) (- 1.0 (cos y))) 2.0)
        (+
         t_1
         (fma (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 0.6666666666666666) 3.0)))
       (/
        (fma (sqrt 2.0) (* t_2 (* -0.0625 (pow (sin x) 2.0))) 2.0)
        (+ t_1 (fma (cos y) (/ t_3 0.6666666666666666) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = (cos(x) * t_0) / 0.6666666666666666;
	double t_2 = cos(x) + -1.0;
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0012) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * t_2)) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_3 / 2.0))));
	} else if (x <= 0.00072) {
		tmp = fma(sqrt(2.0), ((-0.0625 * pow(sin(y), 2.0)) * (1.0 - cos(y))), 2.0) / (t_1 + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) / 0.6666666666666666), 3.0));
	} else {
		tmp = fma(sqrt(2.0), (t_2 * (-0.0625 * pow(sin(x), 2.0))), 2.0) / (t_1 + fma(cos(y), (t_3 / 0.6666666666666666), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(Float64(cos(x) * t_0) / 0.6666666666666666)
	t_2 = Float64(cos(x) + -1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0012)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_2)) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_3 / 2.0)))));
	elseif (x <= 0.00072)
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(-0.0625 * (sin(y) ^ 2.0)) * Float64(1.0 - cos(y))), 2.0) / Float64(t_1 + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 0.6666666666666666), 3.0)));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(t_2 * Float64(-0.0625 * (sin(x) ^ 2.0))), 2.0) / Float64(t_1 + fma(cos(y), Float64(t_3 / 0.6666666666666666), 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[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0012], N[(N[(2.0 + 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$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00072], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$3 / 0.6666666666666666), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00119999999999999989

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 61.5%

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

   3. Taylor expanded in y around 0 60.7%

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

      1. *-commutative 63.3%

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

   5. Simplified 60.7%

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

   if -0.00119999999999999989 < x < 7.20000000000000045e-4

   1. Initial program 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{-0.0625 \cdot \left({\sin y}^{2} \cdot \left(1 - \cos y\right)\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

      2. *-commutative 98.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

   10. Simplified 98.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666}, 3\right)} \]

   if 7.20000000000000045e-4 < x

   1. Initial program 98.9%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 59.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 59.5%

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

      2. *-commutative 59.5%

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

      3. sub-neg 59.5%

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

      4. metadata-eval 59.5%

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

   6. Simplified 59.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0012:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00072:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(-0.0625 \cdot {\sin y}^{2}\right) \cdot \left(1 - \cos y\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x + -1\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}\\ \end{array} \]

Alternative 13: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 1 + \cos x \cdot \frac{t_0}{2}\\ t_2 := \cos x + -1\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0012:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t_2}{3 \cdot \left(t_1 + \cos y \cdot \frac{t_3}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00088:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, t_2 \cdot \left(-0.0625 \cdot {\sin x}^{2}\right), 2\right)}{\frac{\cos x \cdot t_0}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{t_3}{0.6666666666666666}, 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (+ 1.0 (* (cos x) (/ t_0 2.0))))
        (t_2 (+ (cos x) -1.0))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -0.0012)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))) t_2))
      (* 3.0 (+ t_1 (* (cos y) (/ t_3 2.0)))))
     (if (<= x 0.00088)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
        (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
       (/
        (fma (sqrt 2.0) (* t_2 (* -0.0625 (pow (sin x) 2.0))) 2.0)
        (+
         (/ (* (cos x) t_0) 0.6666666666666666)
         (fma (cos y) (/ t_3 0.6666666666666666) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 1.0 + (cos(x) * (t_0 / 2.0));
	double t_2 = cos(x) + -1.0;
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -0.0012) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * t_2)) / (3.0 * (t_1 + (cos(y) * (t_3 / 2.0))));
	} else if (x <= 0.00088) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), (t_2 * (-0.0625 * pow(sin(x), 2.0))), 2.0) / (((cos(x) * t_0) / 0.6666666666666666) + fma(cos(y), (t_3 / 0.6666666666666666), 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0)))
	t_2 = Float64(cos(x) + -1.0)
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -0.0012)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_2)) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(t_3 / 2.0)))));
	elseif (x <= 0.00088)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(t_2 * Float64(-0.0625 * (sin(x) ^ 2.0))), 2.0) / Float64(Float64(Float64(cos(x) * t_0) / 0.6666666666666666) + fma(cos(y), Float64(t_3 / 0.6666666666666666), 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0012], N[(N[(2.0 + 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$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00088], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$3 / 0.6666666666666666), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00119999999999999989

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 61.5%

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

   3. Taylor expanded in y around 0 60.7%

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

      1. *-commutative 63.3%

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

   5. Simplified 60.7%

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

   if -0.00119999999999999989 < x < 8.80000000000000031e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   9. Applied egg-rr 98.3%

      \[\leadsto \frac{2 + \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]
   10. Step-by-step derivation

       1. +-commutative 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   11. Simplified 98.3%

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

   if 8.80000000000000031e-4 < x

   1. Initial program 98.9%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 59.5%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \color{blue}{-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x - 1\right)\right)}, 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 59.5%

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

      2. *-commutative 59.5%

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

      3. sub-neg 59.5%

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

      4. metadata-eval 59.5%

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

   6. Simplified 59.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0012:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00088:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\cos x + -1\right) \cdot \left(-0.0625 \cdot {\sin x}^{2}\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}\\ \end{array} \]

Alternative 14: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ \mathbf{if}\;x \leq -0.0014:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + t_0, \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1
         (+
          2.0
          (* (pow (sin x) 2.0) (* -0.0625 (* (sqrt 2.0) (+ (cos x) -1.0)))))))
   (if (<= x -0.0014)
     (/
      t_1
      (*
       3.0
       (+
        1.0
        (+
         (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5))
         (* (cos y) (/ 1.0 (fma 0.5 (sqrt 5.0) 1.5)))))))
     (if (<= x 0.00125)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
       (/
        t_1
        (* 3.0 (+ 1.0 (fma (cos x) (+ -0.5 t_0) (* (cos y) (- 1.5 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (pow(sin(x), 2.0) * (-0.0625 * (sqrt(2.0) * (cos(x) + -1.0))));
	double tmp;
	if (x <= -0.0014) {
		tmp = t_1 / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (1.0 / fma(0.5, sqrt(5.0), 1.5))))));
	} else if (x <= 0.00125) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = t_1 / (3.0 * (1.0 + fma(cos(x), (-0.5 + t_0), (cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(-0.0625 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	tmp = 0.0
	if (x <= -0.0014)
		tmp = Float64(t_1 / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(1.0 / fma(0.5, sqrt(5.0), 1.5)))))));
	elseif (x <= 0.00125)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(1.0 + fma(cos(x), Float64(-0.5 + t_0), Float64(cos(y) * Float64(1.5 - t_0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0014], N[(t$95$1 / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00125], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00139999999999999999

   1. Initial program 98.7%

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

      1. associate-*l* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. cos-neg 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. fma-def 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 60.4%

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

      1. *-commutative 60.3%

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

      2. sub-neg 60.3%

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

      3. metadata-eval 60.3%

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

      4. associate-*l* 60.3%

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

   10. Simplified 60.4%

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

   if -0.00139999999999999999 < x < 0.00125000000000000003

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   9. Applied egg-rr 98.3%

      \[\leadsto \frac{2 + \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]
   10. Step-by-step derivation

       1. +-commutative 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   11. Simplified 98.3%

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

   if 0.00125000000000000003 < 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. Step-by-step derivation

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

      1. fma-def 99.0%

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

      2. sub-neg 99.0%

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

      3. div-inv 99.0%

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

      4. metadata-eval 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. sub-neg 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*l* 59.5%

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

   8. Simplified 59.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0014:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 15: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x + -1\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.00085:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot t_0\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_2 - 0.5\right) + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.00094:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + t_1, \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (cos x) -1.0))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (/ (sqrt 5.0) 2.0)))
   (if (<= x -0.00085)
     (/
      (+ 2.0 (* (* (sqrt 2.0) (sin x)) (* (- (sin y) (/ (sin x) 16.0)) t_0)))
      (* 3.0 (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))
     (if (<= x 0.00094)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
       (/
        (+ 2.0 (* (pow (sin x) 2.0) (* -0.0625 (* (sqrt 2.0) t_0))))
        (* 3.0 (+ 1.0 (fma (cos x) (+ -0.5 t_1) (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) + -1.0;
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.00085) {
		tmp = (2.0 + ((sqrt(2.0) * sin(x)) * ((sin(y) - (sin(x) / 16.0)) * t_0))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	} else if (x <= 0.00094) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + (pow(sin(x), 2.0) * (-0.0625 * (sqrt(2.0) * t_0)))) / (3.0 * (1.0 + fma(cos(x), (-0.5 + t_1), (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) + -1.0)
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (x <= -0.00085)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * sin(x)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * t_0))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	elseif (x <= 0.00094)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(-0.0625 * Float64(sqrt(2.0) * t_0)))) / Float64(3.0 * Float64(1.0 + fma(cos(x), Float64(-0.5 + t_1), Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.00085], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00094], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.49999999999999953e-4

   1. Initial program 98.7%

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

      1. associate-*l* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. cos-neg 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in y around 0 60.7%

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

   if -8.49999999999999953e-4 < x < 9.39999999999999972e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   9. Applied egg-rr 98.3%

      \[\leadsto \frac{2 + \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]
   10. Step-by-step derivation

       1. +-commutative 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   11. Simplified 98.3%

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

   if 9.39999999999999972e-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. Step-by-step derivation

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

      1. fma-def 99.0%

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

      2. sub-neg 99.0%

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

      3. div-inv 99.0%

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

      4. metadata-eval 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. sub-neg 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*l* 59.5%

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

   8. Simplified 59.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00085:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.00094:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 16: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos x + -1\\ t_2 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ \mathbf{if}\;x \leq -0.00066:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot t_1}{3 \cdot \left(t_2 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot t_1\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + t_0, \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (+ (cos x) -1.0))
        (t_2 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))))
   (if (<= x -0.00066)
     (/
      (+ 2.0 (* (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))) t_1))
      (* 3.0 (+ t_2 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 0.00115)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
        (* 3.0 (+ t_2 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
       (/
        (+ 2.0 (* (pow (sin x) 2.0) (* -0.0625 (* (sqrt 2.0) t_1))))
        (* 3.0 (+ 1.0 (fma (cos x) (+ -0.5 t_0) (* (cos y) (- 1.5 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = cos(x) + -1.0;
	double t_2 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double tmp;
	if (x <= -0.00066) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * t_1)) / (3.0 * (t_2 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 0.00115) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * (t_2 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + (pow(sin(x), 2.0) * (-0.0625 * (sqrt(2.0) * t_1)))) / (3.0 * (1.0 + fma(cos(x), (-0.5 + t_0), (cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(cos(x) + -1.0)
	t_2 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	tmp = 0.0
	if (x <= -0.00066)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * t_1)) / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (x <= 0.00115)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(-0.0625 * Float64(sqrt(2.0) * t_1)))) / Float64(3.0 * Float64(1.0 + fma(cos(x), Float64(-0.5 + t_0), Float64(cos(y) * Float64(1.5 - t_0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00066], N[(N[(2.0 + 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$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00115], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.6e-4

   1. Initial program 98.7%

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

   2. Taylor expanded in y around 0 61.5%

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

   3. Taylor expanded in y around 0 60.7%

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

      1. *-commutative 63.3%

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

   5. Simplified 60.7%

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

   if -6.6e-4 < x < 0.00115

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   9. Applied egg-rr 98.3%

      \[\leadsto \frac{2 + \left({\sin y}^{2} \cdot \left(-0.0625 \cdot \sqrt{2}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)} \]
   10. Step-by-step derivation

       1. +-commutative 99.7%

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   11. Simplified 98.3%

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

   if 0.00115 < 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. Step-by-step derivation

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

      1. fma-def 99.0%

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

      2. sub-neg 99.0%

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

      3. div-inv 99.0%

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

      4. metadata-eval 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. sub-neg 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*l* 59.5%

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

   8. Simplified 59.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00066:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 17: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := 2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{t_2}{t_0}\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + t_1, \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           1.0
           (+
            (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5))
            (* (cos y) (/ 1.0 (fma 0.5 (sqrt 5.0) 1.5)))))))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2
         (+
          2.0
          (* (pow (sin x) 2.0) (* -0.0625 (* (sqrt 2.0) (+ (cos x) -1.0)))))))
   (if (<= x -0.00072)
     (/ t_2 t_0)
     (if (<= x 0.00115)
       (/
        (+
         2.0
         (* (pow (sin y) 2.0) (* -0.0625 (* (sqrt 2.0) (- 1.0 (cos y))))))
        t_0)
       (/
        t_2
        (* 3.0 (+ 1.0 (fma (cos x) (+ -0.5 t_1) (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (1.0 / fma(0.5, sqrt(5.0), 1.5)))));
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = 2.0 + (pow(sin(x), 2.0) * (-0.0625 * (sqrt(2.0) * (cos(x) + -1.0))));
	double tmp;
	if (x <= -0.00072) {
		tmp = t_2 / t_0;
	} else if (x <= 0.00115) {
		tmp = (2.0 + (pow(sin(y), 2.0) * (-0.0625 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else {
		tmp = t_2 / (3.0 * (1.0 + fma(cos(x), (-0.5 + t_1), (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(1.0 / fma(0.5, sqrt(5.0), 1.5))))))
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(-0.0625 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	tmp = 0.0
	if (x <= -0.00072)
		tmp = Float64(t_2 / t_0);
	elseif (x <= 0.00115)
		tmp = Float64(Float64(2.0 + Float64((sin(y) ^ 2.0) * Float64(-0.0625 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	else
		tmp = Float64(t_2 / Float64(3.0 * Float64(1.0 + fma(cos(x), Float64(-0.5 + t_1), Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00072], N[(t$95$2 / t$95$0), $MachinePrecision], If[LessEqual[x, 0.00115], N[(N[(2.0 + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.20000000000000045e-4

   1. Initial program 98.7%

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

      1. associate-*l* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. cos-neg 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. fma-def 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 60.4%

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

      1. *-commutative 60.3%

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

      2. sub-neg 60.3%

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

      3. metadata-eval 60.3%

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

      4. associate-*l* 60.3%

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

   10. Simplified 60.4%

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

   if -7.20000000000000045e-4 < x < 0.00115

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   10. Simplified 98.3%

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

   if 0.00115 < 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. Step-by-step derivation

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

      1. fma-def 99.0%

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

      2. sub-neg 99.0%

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

      3. div-inv 99.0%

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

      4. metadata-eval 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. sub-neg 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*l* 59.5%

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

   8. Simplified 59.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 18: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := -0.5 + t_0\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-5} \lor \neg \left(x \leq 0.00014\right):\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, t_1, \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (+ -0.5 t_0)))
   (if (or (<= x -1.05e-5) (not (<= x 0.00014)))
     (/
      (+ 2.0 (* (pow (sin x) 2.0) (* -0.0625 (* (sqrt 2.0) (+ (cos x) -1.0)))))
      (* 3.0 (+ 1.0 (fma (cos x) t_1 (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
      (* 3.0 (+ (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)) (+ 1.0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = -0.5 + t_0;
	double tmp;
	if ((x <= -1.05e-5) || !(x <= 0.00014)) {
		tmp = (2.0 + (pow(sin(x), 2.0) * (-0.0625 * (sqrt(2.0) * (cos(x) + -1.0))))) / (3.0 * (1.0 + fma(cos(x), t_1, (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + t_1)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(-0.5 + t_0)
	tmp = 0.0
	if ((x <= -1.05e-5) || !(x <= 0.00014))
		tmp = Float64(Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(-0.0625 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(3.0 * Float64(1.0 + fma(cos(x), t_1, Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + t_1))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 + t$95$0), $MachinePrecision]}, If[Or[LessEqual[x, -1.05e-5], N[Not[LessEqual[x, 0.00014]], $MachinePrecision]], N[(N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04999999999999994e-5 or 1.3999999999999999e-4 < x

   1. Initial program 98.8%

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

      1. associate-*l* 98.8%

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

      2. distribute-lft-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-lft-in 98.8%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

      1. fma-def 98.8%

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

      2. sub-neg 98.8%

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

      3. div-inv 98.8%

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

      4. metadata-eval 98.8%

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

      5. metadata-eval 98.8%

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

      6. div-inv 98.8%

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

      7. metadata-eval 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in y around 0 60.0%

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

      1. *-commutative 60.0%

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

      2. sub-neg 60.0%

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

      3. metadata-eval 60.0%

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

      4. associate-*l* 60.0%

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

   8. Simplified 60.0%

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

   if -1.04999999999999994e-5 < x < 1.3999999999999999e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. sub-neg 98.3%

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

      2. metadata-eval 98.3%

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

      3. +-commutative 98.3%

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

      4. distribute-rgt-in 98.3%

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

      5. metadata-eval 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-5} \lor \neg \left(x \leq 0.00014\right):\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 19: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := -0.5 + t_1\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, t_2, \cos y \cdot \left(1.5 - t_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* (pow (sin x) 2.0) (* -0.0625 (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (+ -0.5 t_1)))
   (if (<= x -5.2e-6)
     (/
      t_0
      (*
       3.0
       (+
        1.0
        (+
         (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5))
         (* (cos y) (/ 1.0 (fma 0.5 (sqrt 5.0) 1.5)))))))
     (if (<= x 0.00012)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
        (* 3.0 (+ (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)) (+ 1.0 t_2))))
       (/ t_0 (* 3.0 (+ 1.0 (fma (cos x) t_2 (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (pow(sin(x), 2.0) * (-0.0625 * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = -0.5 + t_1;
	double tmp;
	if (x <= -5.2e-6) {
		tmp = t_0 / (3.0 * (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (1.0 / fma(0.5, sqrt(5.0), 1.5))))));
	} else if (x <= 0.00012) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + t_2)));
	} else {
		tmp = t_0 / (3.0 * (1.0 + fma(cos(x), t_2, (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(-0.0625 * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = Float64(-0.5 + t_1)
	tmp = 0.0
	if (x <= -5.2e-6)
		tmp = Float64(t_0 / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(1.0 / fma(0.5, sqrt(5.0), 1.5)))))));
	elseif (x <= 0.00012)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + t_2))));
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(1.0 + fma(cos(x), t_2, Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 + t$95$1), $MachinePrecision]}, If[LessEqual[x, -5.2e-6], N[(t$95$0 / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00012], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000019e-6

   1. Initial program 98.7%

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

      1. associate-*l* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. cos-neg 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. associate-+l+ 98.8%

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

   3. Simplified 98.8%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. fma-def 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 60.4%

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

      1. *-commutative 60.3%

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

      2. sub-neg 60.3%

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

      3. metadata-eval 60.3%

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

      4. associate-*l* 60.3%

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

   10. Simplified 60.4%

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

   if -5.20000000000000019e-6 < x < 1.20000000000000003e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. sub-neg 98.3%

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

      2. metadata-eval 98.3%

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

      3. +-commutative 98.3%

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

      4. distribute-rgt-in 98.3%

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

      5. metadata-eval 98.3%

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

   10. Simplified 98.3%

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

   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. Step-by-step derivation

      1. associate-*l* 98.9%

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

      2. distribute-lft-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-lft-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

      1. fma-def 99.0%

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

      2. sub-neg 99.0%

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

      3. div-inv 99.0%

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

      4. metadata-eval 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. sub-neg 59.5%

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

      3. metadata-eval 59.5%

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

      4. associate-*l* 59.5%

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

   8. Simplified 59.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{1}{\mathsf{fma}\left(0.5, \sqrt{5}, 1.5\right)}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 \cdot \left(1 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{5} \cdot 0.5, \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 20: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-6} \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(t_0 + \left(1 + \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))
   (if (or (<= x -5.8e-6) (not (<= x 0.00012)))
     (/
      (+ 2.0 (* (+ (cos x) -1.0) (* (sqrt 2.0) (* -0.0625 (pow (sin x) 2.0)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))) t_0)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
      (* 3.0 (+ t_0 (+ 1.0 (+ -0.5 (* (sqrt 5.0) 0.5)))))))))

C

double code(double x, double y) {
	double t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double tmp;
	if ((x <= -5.8e-6) || !(x <= 0.00012)) {
		tmp = (2.0 + ((cos(x) + -1.0) * (sqrt(2.0) * (-0.0625 * pow(sin(x), 2.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + t_0));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * (t_0 + (1.0 + (-0.5 + (sqrt(5.0) * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)
    if ((x <= (-5.8d-6)) .or. (.not. (x <= 0.00012d0))) then
        tmp = (2.0d0 + ((cos(x) + (-1.0d0)) * (sqrt(2.0d0) * ((-0.0625d0) * (sin(x) ** 2.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + t_0))
    else
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (-0.0625d0))))) / (3.0d0 * (t_0 + (1.0d0 + ((-0.5d0) + (sqrt(5.0d0) * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0);
	double tmp;
	if ((x <= -5.8e-6) || !(x <= 0.00012)) {
		tmp = (2.0 + ((Math.cos(x) + -1.0) * (Math.sqrt(2.0) * (-0.0625 * Math.pow(Math.sin(x), 2.0))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + t_0));
	} else {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * -0.0625)))) / (3.0 * (t_0 + (1.0 + (-0.5 + (Math.sqrt(5.0) * 0.5)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)
	tmp = 0
	if (x <= -5.8e-6) or not (x <= 0.00012):
		tmp = (2.0 + ((math.cos(x) + -1.0) * (math.sqrt(2.0) * (-0.0625 * math.pow(math.sin(x), 2.0))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + t_0))
	else:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * -0.0625)))) / (3.0 * (t_0 + (1.0 + (-0.5 + (math.sqrt(5.0) * 0.5)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	tmp = 0.0
	if ((x <= -5.8e-6) || !(x <= 0.00012))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + t_0)));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(t_0 + Float64(1.0 + Float64(-0.5 + Float64(sqrt(5.0) * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	tmp = 0.0;
	if ((x <= -5.8e-6) || ~((x <= 0.00012)))
		tmp = (2.0 + ((cos(x) + -1.0) * (sqrt(2.0) * (-0.0625 * (sin(x) ^ 2.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + t_0));
	else
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) ^ 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * (t_0 + (1.0 + (-0.5 + (sqrt(5.0) * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -5.8e-6], N[Not[LessEqual[x, 0.00012]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(1.0 + N[(-0.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8000000000000004e-6 or 1.20000000000000003e-4 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 60.7%

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

   3. Taylor expanded in y around 0 59.9%

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

      1. associate-*r* 59.9%

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

      2. *-commutative 59.9%

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

   5. Simplified 59.9%

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

   if -5.8000000000000004e-6 < x < 1.20000000000000003e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. sub-neg 98.3%

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

      2. metadata-eval 98.3%

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

      3. +-commutative 98.3%

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

      4. distribute-rgt-in 98.3%

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

      5. metadata-eval 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-6} \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \end{array} \]

Alternative 21: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \frac{1}{1.5 + t_1}\right)}\\ \mathbf{elif}\;x \leq 0.00035:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(-0.5 + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_1 (* (sqrt 5.0) 0.5)))
   (if (<= x -0.029)
     (*
      0.3333333333333333
      (/ t_0 (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (/ 1.0 (+ 1.5 t_1))))))
     (if (<= x 0.00035)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* (pow (sin y) 2.0) (* (sqrt 2.0) -0.0625))))
        (*
         3.0
         (+ (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)) (+ 1.0 (+ -0.5 t_1)))))
       (/
        t_0
        (+
         3.0
         (fma
          1.5
          (* (cos x) (+ (sqrt 5.0) -1.0))
          (/ 6.0 (+ 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_1 = sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -0.029) {
		tmp = 0.3333333333333333 * (t_0 / (1.0 + ((cos(x) * (t_1 - 0.5)) + (1.0 / (1.5 + t_1)))));
	} else if (x <= 0.00035) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (pow(sin(y), 2.0) * (sqrt(2.0) * -0.0625)))) / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + (1.0 + (-0.5 + t_1))));
	} else {
		tmp = t_0 / (3.0 + fma(1.5, (cos(x) * (sqrt(5.0) + -1.0)), (6.0 / (3.0 + sqrt(5.0)))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_1 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if (x <= -0.029)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(1.0 / Float64(1.5 + t_1))))));
	elseif (x <= 0.00035)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + Float64(1.0 + Float64(-0.5 + t_1)))));
	else
		tmp = Float64(t_0 / Float64(3.0 + fma(1.5, Float64(cos(x) * Float64(sqrt(5.0) + -1.0)), Float64(6.0 / Float64(3.0 + sqrt(5.0))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -0.029], N[(0.3333333333333333 * N[(t$95$0 / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00035], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(6.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0290000000000000015

   1. Initial program 98.7%

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

      1. associate-*l* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. cos-neg 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. associate-+l+ 98.7%

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

   3. Simplified 98.7%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. fma-def 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. associate-*l* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*l* 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in y around 0 59.7%

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

   if -0.0290000000000000015 < x < 3.49999999999999996e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 97.7%

      \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
   3. Step-by-step derivation

      1. *-commutative 97.7%

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

      2. associate-*l* 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l* 97.7%

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

      3. *-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0 97.7%

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

      1. sub-neg 97.7%

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

      2. metadata-eval 97.7%

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

      3. +-commutative 97.7%

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

      4. distribute-rgt-in 97.7%

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

      5. metadata-eval 97.7%

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

   10. Simplified 97.7%

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

   if 3.49999999999999996e-4 < x

   1. Initial program 98.9%

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

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 58.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]
   9. Step-by-step derivation

      1. sub-neg 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      2. metadata-eval 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      3. fma-def 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \color{blue}{\mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} - 1\right), 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

      4. sub-neg 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}, 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      5. metadata-eval 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right), 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      6. associate-*r/ 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \color{blue}{\frac{6 \cdot 1}{3 + \sqrt{5}}}\right)} \]

      7. metadata-eval 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{\color{blue}{6}}{3 + \sqrt{5}}\right)} \]

   10. Simplified 58.6%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{elif}\;x \leq 0.00035:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \left(-0.5 + \sqrt{5} \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 22: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_3 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_2}{1 + \left(\cos x \cdot \left(t_3 - 0.5\right) + \frac{1}{1.5 + t_3}\right)}\\ \mathbf{elif}\;x \leq 0.00015:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t_0} + 1.5 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 + \mathsf{fma}\left(1.5, \cos x \cdot t_1, \frac{6}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_3 (* (sqrt 5.0) 0.5)))
   (if (<= x -0.029)
     (*
      0.3333333333333333
      (/ t_2 (+ 1.0 (+ (* (cos x) (- t_3 0.5)) (/ 1.0 (+ 1.5 t_3))))))
     (if (<= x 0.00015)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (+ (* 6.0 (/ (cos y) t_0)) (* 1.5 t_1))))
       (/ t_2 (+ 3.0 (fma 1.5 (* (cos x) t_1) (/ 6.0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_3 = sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -0.029) {
		tmp = 0.3333333333333333 * (t_2 / (1.0 + ((cos(x) * (t_3 - 0.5)) + (1.0 / (1.5 + t_3)))));
	} else if (x <= 0.00015) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	} else {
		tmp = t_2 / (3.0 + fma(1.5, (cos(x) * t_1), (6.0 / t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_3 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if (x <= -0.029)
		tmp = Float64(0.3333333333333333 * Float64(t_2 / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_3 - 0.5)) + Float64(1.0 / Float64(1.5 + t_3))))));
	elseif (x <= 0.00015)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / t_0)) + Float64(1.5 * t_1))));
	else
		tmp = Float64(t_2 / Float64(3.0 + fma(1.5, Float64(cos(x) * t_1), Float64(6.0 / t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -0.029], N[(0.3333333333333333 * N[(t$95$2 / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.5 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00015], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 + N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(6.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0290000000000000015

   1. Initial program 98.7%

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

      1. associate-*l* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. cos-neg 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. associate-+l+ 98.7%

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

   3. Simplified 98.7%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. fma-def 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. associate-*l* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*l* 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in y around 0 59.7%

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

   if -0.0290000000000000015 < x < 1.49999999999999987e-4

   1. Initial program 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\sqrt{5} - 1\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}} \]

   if 1.49999999999999987e-4 < x

   1. Initial program 98.9%

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

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 58.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]
   9. Step-by-step derivation

      1. sub-neg 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      2. metadata-eval 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      3. fma-def 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \color{blue}{\mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} - 1\right), 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]

      4. sub-neg 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}, 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      5. metadata-eval 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right), 6 \cdot \frac{1}{3 + \sqrt{5}}\right)} \]

      6. associate-*r/ 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \color{blue}{\frac{6 \cdot 1}{3 + \sqrt{5}}}\right)} \]

      7. metadata-eval 58.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{\color{blue}{6}}{3 + \sqrt{5}}\right)} \]

   10. Simplified 58.6%

       \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{elif}\;x \leq 0.00015:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \mathsf{fma}\left(1.5, \cos x \cdot \left(\sqrt{5} + -1\right), \frac{6}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 23: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_3 := \cos x \cdot t_1\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_2}{2.5 + \left(t_3 \cdot 0.5 - \sqrt{5} \cdot 0.5\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t_0} + 1.5 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 + \left(1.5 \cdot t_3 + 6 \cdot \frac{1}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_3 (* (cos x) t_1)))
   (if (<= x -0.029)
     (* 0.3333333333333333 (/ t_2 (+ 2.5 (- (* t_3 0.5) (* (sqrt 5.0) 0.5)))))
     (if (<= x 0.00012)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (+ (* 6.0 (/ (cos y) t_0)) (* 1.5 t_1))))
       (/ t_2 (+ 3.0 (+ (* 1.5 t_3) (* 6.0 (/ 1.0 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_3 = cos(x) * t_1;
	double tmp;
	if (x <= -0.029) {
		tmp = 0.3333333333333333 * (t_2 / (2.5 + ((t_3 * 0.5) - (sqrt(5.0) * 0.5))));
	} else if (x <= 0.00012) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	} else {
		tmp = t_2 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    t_2 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    t_3 = cos(x) * t_1
    if (x <= (-0.029d0)) then
        tmp = 0.3333333333333333d0 * (t_2 / (2.5d0 + ((t_3 * 0.5d0) - (sqrt(5.0d0) * 0.5d0))))
    else if (x <= 0.00012d0) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / t_0)) + (1.5d0 * t_1)))
    else
        tmp = t_2 / (3.0d0 + ((1.5d0 * t_3) + (6.0d0 * (1.0d0 / t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double t_2 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double t_3 = Math.cos(x) * t_1;
	double tmp;
	if (x <= -0.029) {
		tmp = 0.3333333333333333 * (t_2 / (2.5 + ((t_3 * 0.5) - (Math.sqrt(5.0) * 0.5))));
	} else if (x <= 0.00012) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / t_0)) + (1.5 * t_1)));
	} else {
		tmp = t_2 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = math.sqrt(5.0) + -1.0
	t_2 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	t_3 = math.cos(x) * t_1
	tmp = 0
	if x <= -0.029:
		tmp = 0.3333333333333333 * (t_2 / (2.5 + ((t_3 * 0.5) - (math.sqrt(5.0) * 0.5))))
	elif x <= 0.00012:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / t_0)) + (1.5 * t_1)))
	else:
		tmp = t_2 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_3 = Float64(cos(x) * t_1)
	tmp = 0.0
	if (x <= -0.029)
		tmp = Float64(0.3333333333333333 * Float64(t_2 / Float64(2.5 + Float64(Float64(t_3 * 0.5) - Float64(sqrt(5.0) * 0.5)))));
	elseif (x <= 0.00012)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / t_0)) + Float64(1.5 * t_1))));
	else
		tmp = Float64(t_2 / Float64(3.0 + Float64(Float64(1.5 * t_3) + Float64(6.0 * Float64(1.0 / t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = sqrt(5.0) + -1.0;
	t_2 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	t_3 = cos(x) * t_1;
	tmp = 0.0;
	if (x <= -0.029)
		tmp = 0.3333333333333333 * (t_2 / (2.5 + ((t_3 * 0.5) - (sqrt(5.0) * 0.5))));
	elseif (x <= 0.00012)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	else
		tmp = t_2 / (3.0 + ((1.5 * t_3) + (6.0 * (1.0 / t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[x, -0.029], N[(0.3333333333333333 * N[(t$95$2 / N[(2.5 + N[(N[(t$95$3 * 0.5), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00012], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 + N[(N[(1.5 * t$95$3), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0290000000000000015

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-*l* 98.7%

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

      3. fma-def 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. cos-neg 98.7%

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

      6. distribute-lft-in 98.7%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 59.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   5. Step-by-step derivation

      1. *-commutative 59.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      2. sub-neg 59.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. metadata-eval 59.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. associate--l+ 59.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

      5. sub-neg 59.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right) - 0.5 \cdot \sqrt{5}\right)} \]

      6. metadata-eval 59.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]

   6. Simplified 59.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   if -0.0290000000000000015 < x < 1.20000000000000003e-4

   1. Initial program 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\sqrt{5} - 1\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}} \]

   if 1.20000000000000003e-4 < x

   1. Initial program 98.9%

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

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 58.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 - \sqrt{5} \cdot 0.5\right)}\\ \mathbf{elif}\;x \leq 0.00012:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 24: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ t_2 := 2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)\\ t_3 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_2}{1 + \left(\cos x \cdot \left(t_3 - 0.5\right) + \frac{1}{1.5 + t_3}\right)}\\ \mathbf{elif}\;x \leq 0.00014:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{t_0} + 1.5 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 + \left(1.5 \cdot \left(\cos x \cdot t_1\right) + 6 \cdot \frac{1}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 3.0 (sqrt 5.0)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0))))))
        (t_3 (* (sqrt 5.0) 0.5)))
   (if (<= x -0.029)
     (*
      0.3333333333333333
      (/ t_2 (+ 1.0 (+ (* (cos x) (- t_3 0.5)) (/ 1.0 (+ 1.5 t_3))))))
     (if (<= x 0.00014)
       (/
        (+
         2.0
         (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
        (+ 3.0 (+ (* 6.0 (/ (cos y) t_0)) (* 1.5 t_1))))
       (/ t_2 (+ 3.0 (+ (* 1.5 (* (cos x) t_1)) (* 6.0 (/ 1.0 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 + sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = 2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	double t_3 = sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -0.029) {
		tmp = 0.3333333333333333 * (t_2 / (1.0 + ((cos(x) * (t_3 - 0.5)) + (1.0 / (1.5 + t_3)))));
	} else if (x <= 0.00014) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	} else {
		tmp = t_2 / (3.0 + ((1.5 * (cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 3.0d0 + sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    t_2 = 2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))
    t_3 = sqrt(5.0d0) * 0.5d0
    if (x <= (-0.029d0)) then
        tmp = 0.3333333333333333d0 * (t_2 / (1.0d0 + ((cos(x) * (t_3 - 0.5d0)) + (1.0d0 / (1.5d0 + t_3)))))
    else if (x <= 0.00014d0) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / t_0)) + (1.5d0 * t_1)))
    else
        tmp = t_2 / (3.0d0 + ((1.5d0 * (cos(x) * t_1)) + (6.0d0 * (1.0d0 / t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + Math.sqrt(5.0);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double t_2 = 2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))));
	double t_3 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if (x <= -0.029) {
		tmp = 0.3333333333333333 * (t_2 / (1.0 + ((Math.cos(x) * (t_3 - 0.5)) + (1.0 / (1.5 + t_3)))));
	} else if (x <= 0.00014) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / t_0)) + (1.5 * t_1)));
	} else {
		tmp = t_2 / (3.0 + ((1.5 * (Math.cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + math.sqrt(5.0)
	t_1 = math.sqrt(5.0) + -1.0
	t_2 = 2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))
	t_3 = math.sqrt(5.0) * 0.5
	tmp = 0
	if x <= -0.029:
		tmp = 0.3333333333333333 * (t_2 / (1.0 + ((math.cos(x) * (t_3 - 0.5)) + (1.0 / (1.5 + t_3)))))
	elif x <= 0.00014:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / t_0)) + (1.5 * t_1)))
	else:
		tmp = t_2 / (3.0 + ((1.5 * (math.cos(x) * t_1)) + (6.0 * (1.0 / t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))))
	t_3 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if (x <= -0.029)
		tmp = Float64(0.3333333333333333 * Float64(t_2 / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_3 - 0.5)) + Float64(1.0 / Float64(1.5 + t_3))))));
	elseif (x <= 0.00014)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / t_0)) + Float64(1.5 * t_1))));
	else
		tmp = Float64(t_2 / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(x) * t_1)) + Float64(6.0 * Float64(1.0 / t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + sqrt(5.0);
	t_1 = sqrt(5.0) + -1.0;
	t_2 = 2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))));
	t_3 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if (x <= -0.029)
		tmp = 0.3333333333333333 * (t_2 / (1.0 + ((cos(x) * (t_3 - 0.5)) + (1.0 / (1.5 + t_3)))));
	elseif (x <= 0.00014)
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / t_0)) + (1.5 * t_1)));
	else
		tmp = t_2 / (3.0 + ((1.5 * (cos(x) * t_1)) + (6.0 * (1.0 / t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -0.029], N[(0.3333333333333333 * N[(t$95$2 / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.5 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00014], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 + N[(N[(1.5 * N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(6.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0290000000000000015

   1. Initial program 98.7%

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

      1. associate-*l* 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. cos-neg 98.7%

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

      4. distribute-lft-in 98.7%

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

      5. associate-+l+ 98.7%

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

   3. Simplified 98.7%

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

      1. flip-- 98.6%

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

      2. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 98.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 98.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. fma-def 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. associate-*l* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*l* 99.0%

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

   10. Simplified 99.0%

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

   11. Taylor expanded in y around 0 59.7%

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

   if -0.0290000000000000015 < x < 1.3999999999999999e-4

   1. Initial program 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\sqrt{5} - 1\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}} \]

   if 1.3999999999999999e-4 < x

   1. Initial program 98.9%

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

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.1%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in y around 0 58.6%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{elif}\;x \leq 0.00014:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{3 + \left(1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 6 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]

Alternative 25: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00035\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 - t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_0 + \frac{\cos y}{1.5 + t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.029) (not (<= x 0.00035)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 2.5 (- (* (* (cos x) (+ (sqrt 5.0) -1.0)) 0.5) t_0))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 0.5 (+ t_0 (/ (cos y) (+ 1.5 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00035)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (2.5 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) - t_0)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + (cos(y) / (1.5 + t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-0.029d0)) .or. (.not. (x <= 0.00035d0))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (2.5d0 + (((cos(x) * (sqrt(5.0d0) + (-1.0d0))) * 0.5d0) - t_0)))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (0.5d0 + (t_0 + (cos(y) / (1.5d0 + t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00035)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (2.5 + (((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) * 0.5) - t_0)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (0.5 + (t_0 + (Math.cos(y) / (1.5 + t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -0.029) or not (x <= 0.00035):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (2.5 + (((math.cos(x) * (math.sqrt(5.0) + -1.0)) * 0.5) - t_0)))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (0.5 + (t_0 + (math.cos(y) / (1.5 + t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -0.029) || !(x <= 0.00035))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(2.5 + Float64(Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) * 0.5) - t_0))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(0.5 + Float64(t_0 + Float64(cos(y) / Float64(1.5 + t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -0.029) || ~((x <= 0.00035)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (2.5 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) - t_0)));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + (cos(y) / (1.5 + t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -0.029], N[Not[LessEqual[x, 0.00035]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] / N[(1.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 3.49999999999999996e-4 < x

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. fma-def 98.8%

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

      4. distribute-lft-in 98.9%

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

      5. cos-neg 98.9%

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

      6. distribute-lft-in 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 59.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   5. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      2. sub-neg 59.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. metadata-eval 59.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. associate--l+ 59.2%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

      5. sub-neg 59.2%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right) - 0.5 \cdot \sqrt{5}\right)} \]

      6. metadata-eval 59.2%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]

   6. Simplified 59.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   if -0.0290000000000000015 < x < 3.49999999999999996e-4

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25} - \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      3. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      5. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)\right)} \]

      7. div-inv 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \color{blue}{\sqrt{5} \cdot \frac{1}{2}}}\right)\right)} \]

      8. metadata-eval 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot \color{blue}{0.5}}\right)\right)} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \color{blue}{\frac{2.25 - \left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}}\right)\right)} \]
   6. Step-by-step derivation

      1. swap-sqr 99.5%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      2. rem-square-sqrt 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 - \color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      3. cancel-sign-sub-inv 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{\color{blue}{2.25 + \left(-5\right) \cdot \left(0.5 \cdot 0.5\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-5} \cdot \left(0.5 \cdot 0.5\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      5. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + -5 \cdot \color{blue}{0.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \sqrt{5} \cdot 0.5}\right)\right)} \]

      7. metadata-eval 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. associate-*l* 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in x around 0 97.5%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00035\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 - \sqrt{5} \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \frac{\cos y}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \end{array} \]

Alternative 26: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(\left(\cos x \cdot t_0\right) \cdot 0.5 - \sqrt{5} \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -0.029) (not (<= x 0.00012)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 2.5 (- (* (* (cos x) t_0) 0.5) (* (sqrt 5.0) 0.5)))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+ 3.0 (+ (* 6.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))) (* 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00012)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (2.5 + (((cos(x) * t_0) * 0.5) - (sqrt(5.0) * 0.5))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-0.029d0)) .or. (.not. (x <= 0.00012d0))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (2.5d0 + (((cos(x) * t_0) * 0.5d0) - (sqrt(5.0d0) * 0.5d0))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + ((6.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))) + (1.5d0 * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -0.029) || !(x <= 0.00012)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (2.5 + (((Math.cos(x) * t_0) * 0.5) - (Math.sqrt(5.0) * 0.5))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + ((6.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))) + (1.5 * t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -0.029) or not (x <= 0.00012):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (2.5 + (((math.cos(x) * t_0) * 0.5) - (math.sqrt(5.0) * 0.5))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + ((6.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))) + (1.5 * t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -0.029) || !(x <= 0.00012))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(2.5 + Float64(Float64(Float64(cos(x) * t_0) * 0.5) - Float64(sqrt(5.0) * 0.5)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(Float64(6.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0)))) + Float64(1.5 * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -0.029) || ~((x <= 0.00012)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (2.5 + (((cos(x) * t_0) * 0.5) - (sqrt(5.0) * 0.5))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + ((6.0 * (cos(y) / (3.0 + sqrt(5.0)))) + (1.5 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.029], N[Not[LessEqual[x, 0.00012]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(N[(6.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 1.20000000000000003e-4 < x

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. fma-def 98.8%

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

      4. distribute-lft-in 98.9%

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

      5. cos-neg 98.9%

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

      6. distribute-lft-in 98.8%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 59.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   5. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      2. sub-neg 59.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. metadata-eval 59.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. associate--l+ 59.2%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

      5. sub-neg 59.2%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right) - 0.5 \cdot \sqrt{5}\right)} \]

      6. metadata-eval 59.2%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]

   6. Simplified 59.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   if -0.0290000000000000015 < x < 1.20000000000000003e-4

   1. Initial program 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, 3\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.6%

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

      2. metadata-eval 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      3. add-sqr-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{5}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, 3\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{3 + \sqrt{5}}}}{0.6666666666666666}, 3\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   7. Simplified 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{\frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666} + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, 3\right)} \]

   8. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(1.5 \cdot \left(\sqrt{5} - 1\right) + 6 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029 \lor \neg \left(x \leq 0.00012\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 - \sqrt{5} \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + \left(6 \cdot \frac{\cos y}{3 + \sqrt{5}} + 1.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 27: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5\right) - \sqrt{5} \cdot 0.5} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
   (- (+ 2.5 (* (* (cos x) (+ (sqrt 5.0) -1.0)) 0.5)) (* (sqrt 5.0) 0.5)))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((2.5 + ((cos(x) * (sqrt(5.0) + -1.0)) * 0.5)) - (sqrt(5.0) * 0.5)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / ((2.5d0 + ((cos(x) * (sqrt(5.0d0) + (-1.0d0))) * 0.5d0)) - (sqrt(5.0d0) * 0.5d0)))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / ((2.5 + ((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) * 0.5)) - (Math.sqrt(5.0) * 0.5)));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / ((2.5 + ((math.cos(x) * (math.sqrt(5.0) + -1.0)) * 0.5)) - (math.sqrt(5.0) * 0.5)))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(2.5 + Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) * 0.5)) - Float64(sqrt(5.0) * 0.5))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / ((2.5 + ((cos(x) * (sqrt(5.0) + -1.0)) * 0.5)) - (sqrt(5.0) * 0.5)));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. +-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. fma-def 99.2%

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

   4. distribute-lft-in 99.2%

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

   5. cos-neg 99.2%

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

   6. distribute-lft-in 99.2%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}}} \]

5. Final simplification 59.0%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(2.5 + \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5\right) - \sqrt{5} \cdot 0.5} \]

Alternative 28: 60.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 - \sqrt{5} \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
   (+ 2.5 (- (* (* (cos x) (+ (sqrt 5.0) -1.0)) 0.5) (* (sqrt 5.0) 0.5))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (2.5 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) - (sqrt(5.0) * 0.5))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (2.5d0 + (((cos(x) * (sqrt(5.0d0) + (-1.0d0))) * 0.5d0) - (sqrt(5.0d0) * 0.5d0))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (2.5 + (((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) * 0.5) - (Math.sqrt(5.0) * 0.5))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (2.5 + (((math.cos(x) * (math.sqrt(5.0) + -1.0)) * 0.5) - (math.sqrt(5.0) * 0.5))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(2.5 + Float64(Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) * 0.5) - Float64(sqrt(5.0) * 0.5)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (2.5 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) - (sqrt(5.0) * 0.5))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. +-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. fma-def 99.2%

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

   4. distribute-lft-in 99.2%

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

   5. cos-neg 99.2%

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

   6. distribute-lft-in 99.2%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}}} \]
5. Step-by-step derivation

   1. *-commutative 59.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)}\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

   2. sub-neg 59.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

   3. metadata-eval 59.0%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + \color{blue}{-1}\right) \cdot \sqrt{2}\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}} \]

   4. associate--l+ 59.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   5. sub-neg 59.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)}\right) - 0.5 \cdot \sqrt{5}\right)} \]

   6. metadata-eval 59.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right)\right) - 0.5 \cdot \sqrt{5}\right)} \]

6. Simplified 59.1%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\left(\cos x + -1\right) \cdot \sqrt{2}\right)\right)}{2.5 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) - 0.5 \cdot \sqrt{5}\right)}} \]

7. Final simplification 59.1%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{2.5 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 - \sqrt{5} \cdot 0.5\right)} \]

Alternative 29: 41.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right) \cdot -0.010416666666666666 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  0.3333333333333333
  (*
   (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))
   -0.010416666666666666)))

C

double code(double x, double y) {
	return 0.3333333333333333 + ((pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))) * -0.010416666666666666);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 + (((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))) * (-0.010416666666666666d0))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 + ((Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))) * -0.010416666666666666);
}

Python

def code(x, y):
	return 0.3333333333333333 + ((math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))) * -0.010416666666666666)

Julia

function code(x, y)
	return Float64(0.3333333333333333 + Float64(Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))) * -0.010416666666666666))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 + (((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))) * -0.010416666666666666);
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.010416666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. +-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. fma-def 99.2%

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

   4. distribute-lft-in 99.2%

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

   5. cos-neg 99.2%

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

   6. distribute-lft-in 99.2%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}}} \]

5. Taylor expanded in x around 0 38.7%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(2.5 + 0.5 \cdot \left(\sqrt{5} - 1\right)\right) - 0.5 \cdot \sqrt{5}}} \]
6. Step-by-step derivation

   1. associate--l+ 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   2. distribute-lft-out-- 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2.5 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) - \sqrt{5}\right)}} \]

   3. sub-neg 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2.5 + 0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} - \sqrt{5}\right)} \]

   4. metadata-eval 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2.5 + 0.5 \cdot \left(\left(\sqrt{5} + \color{blue}{-1}\right) - \sqrt{5}\right)} \]

7. Simplified 38.7%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{2.5 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) - \sqrt{5}\right)}} \]

8. Taylor expanded in x around inf 38.7%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. distribute-lft-in 38.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot 2 + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)} \]

   2. metadata-eval 38.7%

      \[\leadsto \color{blue}{0.3333333333333333} + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right) \]

   3. sub-neg 38.7%

      \[\leadsto 0.3333333333333333 + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)\right) \]

   4. metadata-eval 38.7%

      \[\leadsto 0.3333333333333333 + 0.16666666666666666 \cdot \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)\right) \]

   5. associate-*r* 38.7%

      \[\leadsto 0.3333333333333333 + \color{blue}{\left(0.16666666666666666 \cdot -0.0625\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)} \]

   6. metadata-eval 38.7%

      \[\leadsto 0.3333333333333333 + \color{blue}{-0.010416666666666666} \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right) \]

10. Simplified 38.7%

    \[\leadsto \color{blue}{0.3333333333333333 + -0.010416666666666666 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)} \]

11. Final simplification 38.7%

    \[\leadsto 0.3333333333333333 + \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right) \cdot -0.010416666666666666 \]

Alternative 30: 41.3% accurate, 1139.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

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. Step-by-step derivation

   1. +-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. fma-def 99.2%

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

   4. distribute-lft-in 99.2%

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

   5. cos-neg 99.2%

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

   6. distribute-lft-in 99.2%

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

3. Simplified 99.3%

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

4. Taylor expanded in y around 0 59.0%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} - 1\right)\right)\right) - 0.5 \cdot \sqrt{5}}} \]

5. Taylor expanded in x around 0 38.7%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(2.5 + 0.5 \cdot \left(\sqrt{5} - 1\right)\right) - 0.5 \cdot \sqrt{5}}} \]
6. Step-by-step derivation

   1. associate--l+ 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{2.5 + \left(0.5 \cdot \left(\sqrt{5} - 1\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   2. distribute-lft-out-- 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2.5 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) - \sqrt{5}\right)}} \]

   3. sub-neg 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2.5 + 0.5 \cdot \left(\color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} - \sqrt{5}\right)} \]

   4. metadata-eval 38.7%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{2.5 + 0.5 \cdot \left(\left(\sqrt{5} + \color{blue}{-1}\right) - \sqrt{5}\right)} \]

7. Simplified 38.7%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{2.5 + 0.5 \cdot \left(\left(\sqrt{5} + -1\right) - \sqrt{5}\right)}} \]

8. Taylor expanded in x around 0 38.7%

   \[\leadsto \color{blue}{0.3333333333333333} \]

9. Final simplification 38.7%

   \[\leadsto 0.3333333333333333 \]

Reproduce

herbie shell --seed 2023290 
(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))))))