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

Percentage Accurate: 99.3% → 99.3%

Time: 56.7s

Alternatives: 28

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin x) (* -0.0625 (sin y)))
    (* (+ (sin y) (* (sin x) -0.0625)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (fma
    (cos y)
    (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 0.6666666666666666)
    (/ (* (cos x) (+ (sqrt 5.0) -1.0)) 0.6666666666666666)))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) + (sin(x) * -0.0625)) * (cos(x) - cos(y)))), 2.0) / (3.0 + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) / 0.6666666666666666), ((cos(x) * (sqrt(5.0) + -1.0)) / 0.6666666666666666)));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 0.6666666666666666), Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) / 0.6666666666666666))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

   1. flip-- 99.4%

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

   2. metadata-eval 99.4%

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

   3. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * fma(Float64(3.0 - sqrt(5.0)), cos(y), 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[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around inf 99.4%

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

   1. distribute-lft-out 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-def 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around inf 99.4%

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

   1. distribute-lft-out 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (- (cos x) (cos y))
    (*
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
     (- (sin y) (/ (sin x) 16.0)))))
  (*
   3.0
   (+
    (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
    (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(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]), $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]

Derivation

1. Initial program 99.3%

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

   1. flip-- 99.4%

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

   2. metadata-eval 99.4%

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

   3. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+
    2.0
    (*
     (sqrt 2.0)
     (*
      (- (cos x) (cos y))
      (* (- (sin x) (* (sin y) 0.0625)) (- (sin y) (* (sin x) 0.0625))))))
   (+
    1.0
    (+
     (* (* (cos x) (+ (sqrt 5.0) -1.0)) 0.5)
     (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (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 = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625d0)) * (sin(y) - (sin(x) * 0.0625d0)))))) / (1.0d0 + (((cos(x) * (sqrt(5.0d0) + (-1.0d0))) * 0.5d0) + (2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * ((Math.sin(x) - (Math.sin(y) * 0.0625)) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (1.0 + (((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * ((math.sin(x) - (math.sin(y) * 0.0625)) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (1.0 + (((math.cos(x) * (math.sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(1.0 + Float64(Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) * 0.5) + Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (cos(y) / (3.0 + sqrt(5.0)))))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. flip-- 99.4%

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

   2. metadata-eval 99.4%

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

   3. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. pow1/2 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. pow-prod-up 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around inf 99.2%

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

6. Final simplification 99.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)} \]
7. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (*
    0.3333333333333333
    (/
     (+
      2.0
      (*
       (sqrt 2.0)
       (*
        (- (cos x) (cos y))
        (* (- (sin x) (* (sin y) 0.0625)) (- (sin y) (* (sin x) 0.0625))))))
     (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	return 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) * 0.5d0
    code = 0.3333333333333333d0 * ((2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625d0)) * (sin(y) - (sin(x) * 0.0625d0)))))) / (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	return 0.3333333333333333 * ((2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * ((Math.sin(x) - (Math.sin(y) * 0.0625)) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	return 0.3333333333333333 * ((2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * ((math.sin(x) - (math.sin(y) * 0.0625)) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.3333333333333333 * ((2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * ((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625)))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. distribute-lft-in 99.3%

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

   5. cos-neg 99.3%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 99.2%

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

6. Final simplification 99.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)} \]
7. Add Preprocessing

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \frac{2 + \sqrt{2} \cdot \left(\left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (/
    (+
     2.0
     (*
      (sqrt 2.0)
      (*
       (+ (sin y) (* (sin x) -0.0625))
       (* (+ (sin x) (* -0.0625 (sin y))) (- (cos x) (cos y))))))
    (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	return (2.0 + (sqrt(2.0) * ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (cos(x) - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) / 2.0d0
    code = (2.0d0 + (sqrt(2.0d0) * ((sin(y) + (sin(x) * (-0.0625d0))) * ((sin(x) + ((-0.0625d0) * sin(y))) * (cos(x) - cos(y)))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	return (2.0 + (Math.sqrt(2.0) * ((Math.sin(y) + (Math.sin(x) * -0.0625)) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.cos(x) - Math.cos(y)))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	return (2.0 + (math.sqrt(2.0) * ((math.sin(y) + (math.sin(x) * -0.0625)) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.cos(x) - math.cos(y)))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(cos(x) - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = (2.0 + (sqrt(2.0) * ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (-0.0625 * sin(y))) * (cos(x) - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-lft-in 99.3%

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

3. Simplified 99.3%

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

   1. add-cube-cbrt 99.3%

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

   2. pow3 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in x around inf 99.3%

   \[\leadsto \frac{2 + \color{blue}{{1}^{0.3333333333333333} \cdot \left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - 0.0625 \cdot \sin y\right) \cdot \left(\sin y - 0.0625 \cdot \sin x\right)\right)\right)\right)}}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
8. Step-by-step derivation

   1. pow-base-1 99.3%

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

   2. *-lft-identity 99.3%

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

   3. associate-*r* 99.3%

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

   4. cancel-sign-sub-inv 99.3%

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

   5. metadata-eval 99.3%

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

   6. cancel-sign-sub-inv 99.3%

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

   7. metadata-eval 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 8: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.009 \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.009) (not (<= x 0.017)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
      (*
       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
       (*
        (sqrt 2.0)
        (*
         (* (- (sin x) (* (sin y) 0.0625)) (- (sin y) (* (sin x) 0.0625)))
         (- 1.0 (cos y)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.009) || !(x <= 0.017)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (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 + (sqrt(2.0) * (((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625))) * (1.0 - cos(y))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (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) / 2.0d0
    if ((x <= (-0.009d0)) .or. (.not. (x <= 0.017d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0d0)) * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (((sin(x) - (sin(y) * 0.0625d0)) * (sin(y) - (sin(x) * 0.0625d0))) * (1.0d0 - cos(y))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (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) / 2.0;
	double tmp;
	if ((x <= -0.009) || !(x <= 0.017)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (((Math.sin(x) - (Math.sin(y) * 0.0625)) * (Math.sin(y) - (Math.sin(x) * 0.0625))) * (1.0 - Math.cos(y))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	tmp = 0
	if (x <= -0.009) or not (x <= 0.017):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (((math.sin(x) - (math.sin(y) * 0.0625)) * (math.sin(y) - (math.sin(x) * 0.0625))) * (1.0 - math.cos(y))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.009) || !(x <= 0.017))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))))) / 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(sqrt(2.0) * Float64(Float64(Float64(sin(x) - Float64(sin(y) * 0.0625)) * Float64(sin(y) - Float64(sin(x) * 0.0625))) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if ((x <= -0.009) || ~((x <= 0.017)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (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 + (sqrt(2.0) * (((sin(x) - (sin(y) * 0.0625)) * (sin(y) - (sin(x) * 0.0625))) * (1.0 - cos(y))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (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] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -0.009], N[Not[LessEqual[x, 0.017]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(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[Sqrt[2.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00899999999999999932 or 0.017000000000000001 < x

   1. Initial program 99.0%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

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

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   7. Simplified 63.3%

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

   if -0.00899999999999999932 < x < 0.017000000000000001

   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%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

   6. Taylor expanded in x around 0 99.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.009 \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x - \sin y \cdot 0.0625\right) \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.023 \lor \neg \left(x \leq 0.038\right):\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (- (cos x) (cos y))))
   (if (or (<= x -0.023) (not (<= x 0.038)))
     (/
      (+ 2.0 (* t_2 (* t_0 (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (/
      (+ 2.0 (* t_2 (* t_0 (* (sqrt 2.0) (+ x (* -0.0625 (sin y)))))))
      (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = cos(x) - cos(y);
	double tmp;
	if ((x <= -0.023) || !(x <= 0.038)) {
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	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) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_2 = cos(x) - cos(y)
    if ((x <= (-0.023d0)) .or. (.not. (x <= 0.038d0))) then
        tmp = (2.0d0 + (t_2 * (t_0 * (sqrt(2.0d0) * sin(x))))) / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else
        tmp = (2.0d0 + (t_2 * (t_0 * (sqrt(2.0d0) * (x + ((-0.0625d0) * sin(y))))))) / (3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    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 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_2 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -0.023) || !(x <= 0.038)) {
		tmp = (2.0 + (t_2 * (t_0 * (Math.sqrt(2.0) * Math.sin(x))))) / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + (t_2 * (t_0 * (Math.sqrt(2.0) * (x + (-0.0625 * Math.sin(y))))))) / (3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_2 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -0.023) or not (x <= 0.038):
		tmp = (2.0 + (t_2 * (t_0 * (math.sqrt(2.0) * math.sin(x))))) / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	else:
		tmp = (2.0 + (t_2 * (t_0 * (math.sqrt(2.0) * (x + (-0.0625 * math.sin(y))))))) / (3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -0.023) || !(x <= 0.038))
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * Float64(sqrt(2.0) * Float64(x + Float64(-0.0625 * sin(y))))))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_2 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -0.023) || ~((x <= 0.038)))
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * sin(x))))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	else
		tmp = (2.0 + (t_2 * (t_0 * (sqrt(2.0) * (x + (-0.0625 * sin(y))))))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	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[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.023], N[Not[LessEqual[x, 0.038]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$2 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $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[(2.0 + N[(t$95$2 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.023 or 0.0379999999999999991 < x

   1. Initial program 99.0%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

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

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   7. Simplified 63.3%

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

   if -0.023 < x < 0.0379999999999999991

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. associate-*r* 99.2%

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

      2. metadata-eval 99.2%

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

      3. distribute-rgt-out 99.2%

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

      4. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.023 \lor \neg \left(x \leq 0.038\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \frac{4}{3 + \sqrt{5}}\\ \mathbf{if}\;x \leq -0.0018 \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{t_1}{0.6666666666666666}, \frac{\cos x \cdot t_0}{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (/ 4.0 (+ 3.0 (sqrt 5.0)))))
   (if (or (<= x -0.0018) (not (<= x 0.017)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (- (sin y) (/ (sin x) 16.0)) (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_1 2.0)))))
     (/
      (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+
       3.0
       (fma
        (cos y)
        (/ t_1 0.6666666666666666)
        (/ (* (cos x) t_0) 0.6666666666666666)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 4.0 / (3.0 + sqrt(5.0));
	double tmp;
	if ((x <= -0.0018) || !(x <= 0.017)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + fma(cos(y), (t_1 / 0.6666666666666666), ((cos(x) * t_0) / 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(4.0 / Float64(3.0 + sqrt(5.0)))
	tmp = 0.0
	if ((x <= -0.0018) || !(x <= 0.017))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + fma(cos(y), Float64(t_1 / 0.6666666666666666), Float64(Float64(cos(x) * t_0) / 0.6666666666666666))));
	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[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0018], N[Not[LessEqual[x, 0.017]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(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$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 0.6666666666666666), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0018 or 0.017000000000000001 < x

   1. Initial program 99.0%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

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

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   7. Simplified 63.3%

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

   if -0.0018 < x < 0.017000000000000001

   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 99.7%

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

      1. flip-- 99.7%

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

      2. metadata-eval 99.7%

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

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 98.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0018 \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -0.0009) (not (<= x 0.017)))
     (/
      (+
       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 (* (- 1.0 (cos y)) (pow (sin y) 2.0))) 2.0)
      (+
       3.0
       (fma
        (cos y)
        (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 0.6666666666666666)
        (/ (* (cos x) (+ (sqrt 5.0) -1.0)) 0.6666666666666666)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -0.0009) || !(x <= 0.017)) {
		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 * ((1.0 - cos(y)) * pow(sin(y), 2.0))), 2.0) / (3.0 + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) / 0.6666666666666666), ((cos(x) * (sqrt(5.0) + -1.0)) / 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -0.0009) || !(x <= 0.017))
		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(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))), 2.0) / Float64(3.0 + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 0.6666666666666666), Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) / 0.6666666666666666))));
	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, 0.017]], $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[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999998e-4 or 0.017000000000000001 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      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 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 63.3%

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

      1. *-commutative 63.3%

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

   7. Simplified 63.3%

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

   if -8.9999999999999998e-4 < x < 0.017000000000000001

   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 99.7%

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

      1. flip-- 99.7%

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

      2. metadata-eval 99.7%

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

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 98.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0009 \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{3 + \sqrt{5}}\\ t_1 := {\sin y}^{2}\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t_1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_2}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right), 2\right)}{2.5 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \sqrt{5} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot t_1\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{t_0}{0.6666666666666666}, \frac{\cos x \cdot t_2}{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 4.0 (+ 3.0 (sqrt 5.0))))
        (t_1 (pow (sin y) 2.0))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= y -5.4e-5)
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 t_1))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_2 2.0))) (* (cos y) (/ t_0 2.0)))))
     (if (<= y 1.85e-11)
       (/
        (*
         0.3333333333333333
         (fma
          -0.0625
          (* (* (sqrt 2.0) (pow (sin x) 2.0)) (+ (cos x) -1.0))
          2.0))
        (+ 2.5 (fma (cos x) (fma 0.5 (sqrt 5.0) -0.5) (* (sqrt 5.0) -0.5))))
       (/
        (fma (sqrt 2.0) (* -0.0625 (* (- 1.0 (cos y)) t_1)) 2.0)
        (+
         3.0
         (fma
          (cos y)
          (/ t_0 0.6666666666666666)
          (/ (* (cos x) t_2) 0.6666666666666666))))))))

C

double code(double x, double y) {
	double t_0 = 4.0 / (3.0 + sqrt(5.0));
	double t_1 = pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (y <= -5.4e-5) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * t_1)))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else if (y <= 1.85e-11) {
		tmp = (0.3333333333333333 * fma(-0.0625, ((sqrt(2.0) * pow(sin(x), 2.0)) * (cos(x) + -1.0)), 2.0)) / (2.5 + fma(cos(x), fma(0.5, sqrt(5.0), -0.5), (sqrt(5.0) * -0.5)));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * ((1.0 - cos(y)) * t_1)), 2.0) / (3.0 + fma(cos(y), (t_0 / 0.6666666666666666), ((cos(x) * t_2) / 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(4.0 / Float64(3.0 + sqrt(5.0)))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (y <= -5.4e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * t_1)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (y <= 1.85e-11)
		tmp = Float64(Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * Float64(cos(x) + -1.0)), 2.0)) / Float64(2.5 + fma(cos(x), fma(0.5, sqrt(5.0), -0.5), Float64(sqrt(5.0) * -0.5))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(Float64(1.0 - cos(y)) * t_1)), 2.0) / Float64(3.0 + fma(cos(y), Float64(t_0 / 0.6666666666666666), Float64(Float64(cos(x) * t_2) / 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y, -5.4e-5], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e-11], N[(N[(0.3333333333333333 * N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 0.6666666666666666), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.3999999999999998e-5

   1. Initial program 99.0%

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

      1. flip-- 99.2%

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

      2. metadata-eval 99.2%

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

      3. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

      8. metadata-eval 99.3%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0 58.2%

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

      1. associate-*r* 58.2%

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

   7. Simplified 58.2%

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

   if -5.3999999999999998e-5 < y < 1.8500000000000001e-11

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. cos-neg 99.5%

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

      6. distribute-lft-in 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]

      2. +-commutative 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2\right)}}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. fma-def 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. *-commutative 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      5. *-commutative 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      6. associate-*l* 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\cos x - 1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)}, 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      7. sub-neg 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      8. metadata-eval 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + \color{blue}{-1}\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      9. associate--l+ 99.0%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{\color{blue}{2.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   7. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{2.5 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), -0.5 \cdot \sqrt{5}\right)}} \]

   if 1.8500000000000001e-11 < y

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

      1. flip-- 99.2%

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

      2. metadata-eval 99.2%

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

      3. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around 0 59.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right), 2\right)}{2.5 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \sqrt{5} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right), 2\right)}{2.5 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \sqrt{5} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-5)
   (/
    (+
     2.0
     (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))))
    (*
     3.0
     (+
      (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
      (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
   (if (<= y 1.85e-11)
     (/
      (*
       0.3333333333333333
       (fma -0.0625 (* (* (sqrt 2.0) (pow (sin x) 2.0)) (+ (cos x) -1.0)) 2.0))
      (+ 2.5 (fma (cos x) (fma 0.5 (sqrt 5.0) -0.5) (* (sqrt 5.0) -0.5))))
     (/
      (+
       2.0
       (*
        (* (sqrt 2.0) (* -0.0625 (sin y)))
        (* (- (sin y) (/ (sin x) 16.0)) (- 1.0 (cos y)))))
      (*
       3.0
       (+
        (* (cos y) (- 1.5 (/ (sqrt 5.0) 2.0)))
        (+ 1.0 (/ (cos x) (+ 0.5 (sqrt 1.25))))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-5) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))))) / (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 if (y <= 1.85e-11) {
		tmp = (0.3333333333333333 * fma(-0.0625, ((sqrt(2.0) * pow(sin(x), 2.0)) * (cos(x) + -1.0)), 2.0)) / (2.5 + fma(cos(x), fma(0.5, sqrt(5.0), -0.5), (sqrt(5.0) * -0.5)));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * ((cos(y) * (1.5 - (sqrt(5.0) / 2.0))) + (1.0 + (cos(x) / (0.5 + sqrt(1.25))))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (y <= 1.85e-11)
		tmp = Float64(Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(sqrt(2.0) * (sin(x) ^ 2.0)) * Float64(cos(x) + -1.0)), 2.0)) / Float64(2.5 + fma(cos(x), fma(0.5, sqrt(5.0), -0.5), Float64(sqrt(5.0) * -0.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) / 2.0))) + Float64(1.0 + Float64(cos(x) / Float64(0.5 + sqrt(1.25)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-5], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $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] + 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], If[LessEqual[y, 1.85e-11], N[(N[(0.3333333333333333 * N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.5 + N[(N[Cos[x], $MachinePrecision] * N[(0.5 * N[Sqrt[5.0], $MachinePrecision] + -0.5), $MachinePrecision] + N[(N[Sqrt[5.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[Cos[x], $MachinePrecision] / N[(0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.00000000000000024e-5

   1. Initial program 99.0%

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

      1. flip-- 99.2%

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

      2. metadata-eval 99.2%

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

      3. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.2%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

      8. metadata-eval 99.3%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0 58.2%

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

      1. associate-*r* 58.2%

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

   7. Simplified 58.2%

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

   if -5.00000000000000024e-5 < y < 1.8500000000000001e-11

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. fma-def 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. cos-neg 99.5%

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

      6. distribute-lft-in 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]

      2. +-commutative 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + 2\right)}}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      3. fma-def 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(-0.0625, {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right), 2\right)}}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      4. *-commutative 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\sqrt{2} \cdot \left(\cos x - 1\right)\right) \cdot {\sin x}^{2}}, 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      5. *-commutative 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\left(\cos x - 1\right) \cdot \sqrt{2}\right)} \cdot {\sin x}^{2}, 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      6. associate-*l* 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\cos x - 1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)}, 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      7. sub-neg 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \color{blue}{\left(\cos x + \left(-1\right)\right)} \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      8. metadata-eval 98.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + \color{blue}{-1}\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \]

      9. associate--l+ 99.0%

         \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{\color{blue}{2.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) - 0.5 \cdot \sqrt{5}\right)}} \]

   7. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{2.5 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), -0.5 \cdot \sqrt{5}\right)}} \]

   if 1.8500000000000001e-11 < y

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 58.9%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-*l* 58.9%

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

      4. metadata-eval 58.9%

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

      5. distribute-rgt-neg-in 58.9%

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

      6. *-commutative 58.9%

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

      7. distribute-lft-neg-in 58.9%

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

      8. metadata-eval 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in x around 0 58.9%

      \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\left(1 - \cos y\right)}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
   9. Step-by-step derivation

      1. flip-- 58.9%

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

      2. sub-neg 58.9%

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

      3. frac-times 58.9%

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

      4. pow1/2 58.9%

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

      5. pow1/2 58.9%

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

      6. pow-prod-up 58.9%

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

      7. metadata-eval 58.9%

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

      8. metadata-eval 58.9%

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

      9. metadata-eval 58.9%

         \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\frac{5}{\color{blue}{4}} + \left(-0.5 \cdot 0.5\right)}{\frac{\sqrt{5}}{2} + 0.5}\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]

      10. metadata-eval 58.9%

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\color{blue}{1.25} + \left(-0.5 \cdot 0.5\right)}{\frac{\sqrt{5}}{2} + 0.5}\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]

      11. metadata-eval 58.9%

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{1.25 + \left(-\color{blue}{0.25}\right)}{\frac{\sqrt{5}}{2} + 0.5}\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]

      12. metadata-eval 58.9%

          \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{1.25 + \color{blue}{-0.25}}{\frac{\sqrt{5}}{2} + 0.5}\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]

      13. metadata-eval 58.9%

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

      14. associate-*r/ 58.9%

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

      15. *-rgt-identity 58.9%

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

   10. Applied egg-rr 58.9%

       \[\leadsto \frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\left(1 + \color{blue}{\frac{\cos x}{\sqrt{1.25} + 0.5}}\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\sqrt{2} \cdot {\sin x}^{2}\right) \cdot \left(\cos x + -1\right), 2\right)}{2.5 + \mathsf{fma}\left(\cos x, \mathsf{fma}\left(0.5, \sqrt{5}, -0.5\right), \sqrt{5} \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + \frac{\cos x}{0.5 + \sqrt{1.25}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := t_0 - 0.5\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (- t_0 0.5)) (t_2 (pow (sin x) 2.0)))
   (if (<= x -3.5e-6)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* t_2 (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
       (+ 1.0 (+ (* (cos x) t_1) (* (cos y) (- 1.5 t_0))))))
     (if (<= x 0.017)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (sin y)))
          (* (- (sin y) (/ (sin x) 16.0)) (- 1.0 (cos y)))))
        (* 3.0 (+ (* (cos y) (- 1.5 (/ (sqrt 5.0) 2.0))) (+ 1.0 t_1))))
       (/
        (+ 2.0 (* (- (cos x) (cos y)) (* t_2 (* (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)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = t_0 - 0.5;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -3.5e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * t_1) + (cos(y) * (1.5 - t_0)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * ((cos(y) * (1.5 - (sqrt(5.0) / 2.0))) + (1.0 + t_1)));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_2 * (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))));
	}
	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 = t_0 - 0.5d0
    t_2 = sin(x) ** 2.0d0
    if (x <= (-3.5d-6)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (t_2 * (sqrt(2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))) / (1.0d0 + ((cos(x) * t_1) + (cos(y) * (1.5d0 - t_0)))))
    else if (x <= 0.017d0) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * ((-0.0625d0) * sin(y))) * ((sin(y) - (sin(x) / 16.0d0)) * (1.0d0 - cos(y))))) / (3.0d0 * ((cos(y) * (1.5d0 - (sqrt(5.0d0) / 2.0d0))) + (1.0d0 + t_1)))
    else
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0d0) * (-0.0625d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    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 = t_0 - 0.5;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -3.5e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (Math.sqrt(2.0) * (0.0625 + (-0.0625 * Math.cos(x)))))) / (1.0 + ((Math.cos(x) * t_1) + (Math.cos(y) * (1.5 - t_0)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (-0.0625 * Math.sin(y))) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (1.0 - Math.cos(y))))) / (3.0 * ((Math.cos(y) * (1.5 - (Math.sqrt(5.0) / 2.0))) + (1.0 + t_1)));
	} else {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (t_2 * (Math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = t_0 - 0.5
	t_2 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -3.5e-6:
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (math.sqrt(2.0) * (0.0625 + (-0.0625 * math.cos(x)))))) / (1.0 + ((math.cos(x) * t_1) + (math.cos(y) * (1.5 - t_0)))))
	elif x <= 0.017:
		tmp = (2.0 + ((math.sqrt(2.0) * (-0.0625 * math.sin(y))) * ((math.sin(y) - (math.sin(x) / 16.0)) * (1.0 - math.cos(y))))) / (3.0 * ((math.cos(y) * (1.5 - (math.sqrt(5.0) / 2.0))) + (1.0 + t_1)))
	else:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (t_2 * (math.sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(t_0 - 0.5)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -3.5e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(t_2 * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(1.0 + Float64(Float64(cos(x) * t_1) + Float64(cos(y) * Float64(1.5 - t_0))))));
	elseif (x <= 0.017)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) / 2.0))) + Float64(1.0 + t_1))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_2 * 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)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = t_0 - 0.5;
	t_2 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -3.5e-6)
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * t_1) + (cos(y) * (1.5 - t_0)))));
	elseif (x <= 0.017)
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * ((cos(y) * (1.5 - (sqrt(5.0) / 2.0))) + (1.0 + t_1)));
	else
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_2 * (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))));
	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[(t$95$0 - 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -3.5e-6], N[(0.3333333333333333 * N[(N[(2.0 + N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.017], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * 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]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999995e-6

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.4%

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

      3. associate-*r* 58.4%

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

      4. *-commutative 58.4%

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

      5. sub-neg 58.4%

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

      6. metadata-eval 58.4%

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

      7. distribute-rgt-in 58.4%

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

      8. metadata-eval 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in x around inf 58.5%

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

   if -3.49999999999999995e-6 < x < 0.017000000000000001

   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%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*l* 98.8%

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

      4. metadata-eval 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. distribute-lft-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

   9. Taylor expanded in x around 0 98.8%

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

   if 0.017000000000000001 < x

   1. Initial program 99.0%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

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

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*r* 62.0%

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

   7. Simplified 62.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} \cdot 0.5\\ t_2 := {\sin x}^{2}\\ t_3 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(t_1 - 0.5\right) + \cos y \cdot \left(1.5 - t_1\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* (sqrt 5.0) 0.5))
        (t_2 (pow (sin x) 2.0))
        (t_3
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))
   (if (<= x -0.00065)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* t_2 (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
       (+ 1.0 (+ (* (cos x) (- t_1 0.5)) (* (cos y) (- 1.5 t_1))))))
     (if (<= x 0.017)
       (/ (+ 2.0 (* t_0 (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0))))) t_3)
       (/ (+ 2.0 (* t_0 (* t_2 (* (sqrt 2.0) -0.0625)))) t_3)))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) * 0.5;
	double t_2 = pow(sin(x), 2.0);
	double t_3 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	double tmp;
	if (x <= -0.00065) {
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + (t_0 * (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))))) / t_3;
	} else {
		tmp = (2.0 + (t_0 * (t_2 * (sqrt(2.0) * -0.0625)))) / t_3;
	}
	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 = cos(x) - cos(y)
    t_1 = sqrt(5.0d0) * 0.5d0
    t_2 = sin(x) ** 2.0d0
    t_3 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0)))
    if (x <= (-0.00065d0)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (t_2 * (sqrt(2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))) / (1.0d0 + ((cos(x) * (t_1 - 0.5d0)) + (cos(y) * (1.5d0 - t_1)))))
    else if (x <= 0.017d0) then
        tmp = (2.0d0 + (t_0 * (sqrt(2.0d0) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / t_3
    else
        tmp = (2.0d0 + (t_0 * (t_2 * (sqrt(2.0d0) * (-0.0625d0))))) / t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = Math.sqrt(5.0) * 0.5;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double t_3 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0)));
	double tmp;
	if (x <= -0.00065) {
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (Math.sqrt(2.0) * (0.0625 + (-0.0625 * Math.cos(x)))))) / (1.0 + ((Math.cos(x) * (t_1 - 0.5)) + (Math.cos(y) * (1.5 - t_1)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + (t_0 * (Math.sqrt(2.0) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / t_3;
	} else {
		tmp = (2.0 + (t_0 * (t_2 * (Math.sqrt(2.0) * -0.0625)))) / t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = math.sqrt(5.0) * 0.5
	t_2 = math.pow(math.sin(x), 2.0)
	t_3 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0)))
	tmp = 0
	if x <= -0.00065:
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (math.sqrt(2.0) * (0.0625 + (-0.0625 * math.cos(x)))))) / (1.0 + ((math.cos(x) * (t_1 - 0.5)) + (math.cos(y) * (1.5 - t_1)))))
	elif x <= 0.017:
		tmp = (2.0 + (t_0 * (math.sqrt(2.0) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / t_3
	else:
		tmp = (2.0 + (t_0 * (t_2 * (math.sqrt(2.0) * -0.0625)))) / t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) * 0.5)
	t_2 = sin(x) ^ 2.0
	t_3 = 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))))
	tmp = 0.0
	if (x <= -0.00065)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(t_2 * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_1 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_1))))));
	elseif (x <= 0.017)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / t_3);
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(t_2 * Float64(sqrt(2.0) * -0.0625)))) / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = sqrt(5.0) * 0.5;
	t_2 = sin(x) ^ 2.0;
	t_3 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	tmp = 0.0;
	if (x <= -0.00065)
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * (t_1 - 0.5)) + (cos(y) * (1.5 - t_1)))));
	elseif (x <= 0.017)
		tmp = (2.0 + (t_0 * (sqrt(2.0) * (-0.0625 * (sin(y) ^ 2.0))))) / t_3;
	else
		tmp = (2.0 + (t_0 * (t_2 * (sqrt(2.0) * -0.0625)))) / t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = 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]}, If[LessEqual[x, -0.00065], N[(0.3333333333333333 * N[(N[(2.0 + N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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], If[LessEqual[x, 0.017], N[(N[(2.0 + N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(2.0 + N[(t$95$0 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4999999999999997e-4

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-*l* 57.7%

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

      3. associate-*r* 57.7%

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

      4. *-commutative 57.7%

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

      5. sub-neg 57.7%

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

      6. metadata-eval 57.7%

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

      7. distribute-rgt-in 57.7%

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

      8. metadata-eval 57.7%

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

   7. Simplified 57.7%

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

   8. Taylor expanded in x around inf 57.8%

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

   if -6.4999999999999997e-4 < x < 0.017000000000000001

   1. Initial program 99.6%

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

      1. flip-- 99.7%

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

      2. metadata-eval 99.7%

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

      3. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   7. Simplified 98.8%

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

   if 0.017000000000000001 < x

   1. Initial program 99.0%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

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

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*r* 62.0%

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

   7. Simplified 62.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00065:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := t_0 - 0.5\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (- t_0 0.5)) (t_2 (pow (sin x) 2.0)))
   (if (<= x -2.6e-6)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* t_2 (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
       (+ 1.0 (+ (* (cos x) t_1) (* (cos y) (- 1.5 t_0))))))
     (if (<= x 0.017)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (sin y)))
          (* (- (sin y) (/ (sin x) 16.0)) (- 1.0 (cos y)))))
        (* 3.0 (+ (* (cos y) (- 1.5 (/ (sqrt 5.0) 2.0))) (+ 1.0 t_1))))
       (/
        (+ 2.0 (* (- (cos x) (cos y)) (* t_2 (* (sqrt 2.0) -0.0625))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = t_0 - 0.5;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -2.6e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * t_1) + (cos(y) * (1.5 - t_0)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * ((cos(y) * (1.5 - (sqrt(5.0) / 2.0))) + (1.0 + t_1)));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	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) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = t_0 - 0.5d0
    t_2 = sin(x) ** 2.0d0
    if (x <= (-2.6d-6)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + (t_2 * (sqrt(2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))) / (1.0d0 + ((cos(x) * t_1) + (cos(y) * (1.5d0 - t_0)))))
    else if (x <= 0.017d0) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * ((-0.0625d0) * sin(y))) * ((sin(y) - (sin(x) / 16.0d0)) * (1.0d0 - cos(y))))) / (3.0d0 * ((cos(y) * (1.5d0 - (sqrt(5.0d0) / 2.0d0))) + (1.0d0 + t_1)))
    else
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0d0) * (-0.0625d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    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 = t_0 - 0.5;
	double t_2 = Math.pow(Math.sin(x), 2.0);
	double tmp;
	if (x <= -2.6e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (Math.sqrt(2.0) * (0.0625 + (-0.0625 * Math.cos(x)))))) / (1.0 + ((Math.cos(x) * t_1) + (Math.cos(y) * (1.5 - t_0)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (-0.0625 * Math.sin(y))) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (1.0 - Math.cos(y))))) / (3.0 * ((Math.cos(y) * (1.5 - (Math.sqrt(5.0) / 2.0))) + (1.0 + t_1)));
	} else {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (t_2 * (Math.sqrt(2.0) * -0.0625)))) / (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))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = t_0 - 0.5
	t_2 = math.pow(math.sin(x), 2.0)
	tmp = 0
	if x <= -2.6e-6:
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (math.sqrt(2.0) * (0.0625 + (-0.0625 * math.cos(x)))))) / (1.0 + ((math.cos(x) * t_1) + (math.cos(y) * (1.5 - t_0)))))
	elif x <= 0.017:
		tmp = (2.0 + ((math.sqrt(2.0) * (-0.0625 * math.sin(y))) * ((math.sin(y) - (math.sin(x) / 16.0)) * (1.0 - math.cos(y))))) / (3.0 * ((math.cos(y) * (1.5 - (math.sqrt(5.0) / 2.0))) + (1.0 + t_1)))
	else:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (t_2 * (math.sqrt(2.0) * -0.0625)))) / (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))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(t_0 - 0.5)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -2.6e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(t_2 * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(1.0 + Float64(Float64(cos(x) * t_1) + Float64(cos(y) * Float64(1.5 - t_0))))));
	elseif (x <= 0.017)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) / 2.0))) + Float64(1.0 + t_1))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_2 * 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(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = t_0 - 0.5;
	t_2 = sin(x) ^ 2.0;
	tmp = 0.0;
	if (x <= -2.6e-6)
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * t_1) + (cos(y) * (1.5 - t_0)))));
	elseif (x <= 0.017)
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * ((cos(y) * (1.5 - (sqrt(5.0) / 2.0))) + (1.0 + t_1)));
	else
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	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[(t$95$0 - 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -2.6e-6], N[(0.3333333333333333 * N[(N[(2.0 + N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.017], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * 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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.60000000000000009e-6

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.4%

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

      3. associate-*r* 58.4%

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

      4. *-commutative 58.4%

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

      5. sub-neg 58.4%

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

      6. metadata-eval 58.4%

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

      7. distribute-rgt-in 58.4%

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

      8. metadata-eval 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in x around inf 58.5%

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

   if -2.60000000000000009e-6 < x < 0.017000000000000001

   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%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*l* 98.8%

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

      4. metadata-eval 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. distribute-lft-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

   9. Taylor expanded in x around 0 98.8%

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

   if 0.017000000000000001 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-*r* 62.0%

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

   5. Simplified 62.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := {\sin y}^{2}\\ t_2 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t_1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\left(\cos x \cdot t_0\right) \cdot 0.5 + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_1 \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_2 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (pow (sin y) 2.0))
        (t_2 (/ (sqrt 5.0) 2.0)))
   (if (<= y -5.4e-5)
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 t_1))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= y 1.85e-11)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
         (+
          1.0
          (+ (* (* (cos x) t_0) 0.5) (* 2.0 (/ 1.0 (+ 3.0 (sqrt 5.0))))))))
       (/
        (+ 2.0 (* t_1 (* (sqrt 2.0) (* -0.0625 (- 1.0 (cos y))))))
        (*
         3.0
         (+ (+ 1.0 (* (cos x) (- t_2 0.5))) (* (cos y) (- 1.5 t_2)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) / 2.0;
	double tmp;
	if (y <= -5.4e-5) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * t_1)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (y <= 1.85e-11) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (((cos(x) * t_0) * 0.5) + (2.0 * (1.0 / (3.0 + sqrt(5.0)))))));
	} else {
		tmp = (2.0 + (t_1 * (sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * (t_2 - 0.5))) + (cos(y) * (1.5 - t_2))));
	}
	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) + (-1.0d0)
    t_1 = sin(y) ** 2.0d0
    t_2 = sqrt(5.0d0) / 2.0d0
    if (y <= (-5.4d-5)) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (sqrt(2.0d0) * ((-0.0625d0) * t_1)))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else if (y <= 1.85d-11) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + (((cos(x) * t_0) * 0.5d0) + (2.0d0 * (1.0d0 / (3.0d0 + sqrt(5.0d0)))))))
    else
        tmp = (2.0d0 + (t_1 * (sqrt(2.0d0) * ((-0.0625d0) * (1.0d0 - cos(y)))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_2 - 0.5d0))) + (cos(y) * (1.5d0 - t_2))))
    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 t_1 = Math.pow(Math.sin(y), 2.0);
	double t_2 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if (y <= -5.4e-5) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.sqrt(2.0) * (-0.0625 * t_1)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else if (y <= 1.85e-11) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + (((Math.cos(x) * t_0) * 0.5) + (2.0 * (1.0 / (3.0 + Math.sqrt(5.0)))))));
	} else {
		tmp = (2.0 + (t_1 * (Math.sqrt(2.0) * (-0.0625 * (1.0 - Math.cos(y)))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_2 - 0.5))) + (Math.cos(y) * (1.5 - t_2))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = math.pow(math.sin(y), 2.0)
	t_2 = math.sqrt(5.0) / 2.0
	tmp = 0
	if y <= -5.4e-5:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (math.sqrt(2.0) * (-0.0625 * t_1)))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	elif y <= 1.85e-11:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + (((math.cos(x) * t_0) * 0.5) + (2.0 * (1.0 / (3.0 + math.sqrt(5.0)))))))
	else:
		tmp = (2.0 + (t_1 * (math.sqrt(2.0) * (-0.0625 * (1.0 - math.cos(y)))))) / (3.0 * ((1.0 + (math.cos(x) * (t_2 - 0.5))) + (math.cos(y) * (1.5 - t_2))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (y <= -5.4e-5)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * t_1)))) / 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)))));
	elseif (y <= 1.85e-11)
		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(1.0 + Float64(Float64(Float64(cos(x) * t_0) * 0.5) + Float64(2.0 * Float64(1.0 / Float64(3.0 + sqrt(5.0))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(sqrt(2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = sin(y) ^ 2.0;
	t_2 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if (y <= -5.4e-5)
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * t_1)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	elseif (y <= 1.85e-11)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + (((cos(x) * t_0) * 0.5) + (2.0 * (1.0 / (3.0 + sqrt(5.0)))))));
	else
		tmp = (2.0 + (t_1 * (sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * (t_2 - 0.5))) + (cos(y) * (1.5 - t_2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[y, -5.4e-5], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * t$95$1), $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], If[LessEqual[y, 1.85e-11], 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[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision] + N[(2.0 * N[(1.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.3999999999999998e-5

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. associate-*r* 58.2%

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

   5. Simplified 58.2%

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

   if -5.3999999999999998e-5 < y < 1.8500000000000001e-11

   1. Initial program 99.5%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0 99.0%

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

   if 1.8500000000000001e-11 < y

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 58.9%

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

      3. associate-*r* 58.9%

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

   7. Simplified 58.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := t_0 - 0.5\\ t_2 := {\sin x}^{2}\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + t_2 \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left(t_2 \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (- t_0 0.5)) (t_2 (pow (sin x) 2.0)))
   (if (<= x -5.9e-6)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* t_2 (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
       (+ 1.0 (+ (* (cos x) t_1) (* (cos y) (- 1.5 t_0))))))
     (if (<= x 0.017)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (sin y)))
          (* (- (sin y) (/ (sin x) 16.0)) (- 1.0 (cos y)))))
        (* 3.0 (+ (* (cos y) (- 1.5 (/ (sqrt 5.0) 2.0))) (+ 1.0 t_1))))
       (/
        (fma (sqrt 2.0) (* -0.0625 (* t_2 (+ (cos x) -1.0))) 2.0)
        (+
         3.0
         (*
          1.5
          (+
           (* (cos x) (+ (sqrt 5.0) -1.0))
           (* (cos y) (- 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = t_0 - 0.5;
	double t_2 = pow(sin(x), 2.0);
	double tmp;
	if (x <= -5.9e-6) {
		tmp = 0.3333333333333333 * ((2.0 + (t_2 * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * t_1) + (cos(y) * (1.5 - t_0)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * ((cos(y) * (1.5 - (sqrt(5.0) / 2.0))) + (1.0 + t_1)));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * (t_2 * (cos(x) + -1.0))), 2.0) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(t_0 - 0.5)
	t_2 = sin(x) ^ 2.0
	tmp = 0.0
	if (x <= -5.9e-6)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(t_2 * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(1.0 + Float64(Float64(cos(x) * t_1) + Float64(cos(y) * Float64(1.5 - t_0))))));
	elseif (x <= 0.017)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(1.5 - Float64(sqrt(5.0) / 2.0))) + Float64(1.0 + t_1))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64(t_2 * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	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[(t$95$0 - 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x, -5.9e-6], N[(0.3333333333333333 * N[(N[(2.0 + N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.017], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(t$95$2 * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.90000000000000026e-6

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.4%

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

      3. associate-*r* 58.4%

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

      4. *-commutative 58.4%

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

      5. sub-neg 58.4%

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

      6. metadata-eval 58.4%

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

      7. distribute-rgt-in 58.4%

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

      8. metadata-eval 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in x around inf 58.5%

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

   if -5.90000000000000026e-6 < x < 0.017000000000000001

   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%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*l* 98.8%

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

      4. metadata-eval 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. distribute-lft-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

   9. Taylor expanded in x around 0 98.8%

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

   if 0.017000000000000001 < x

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. distribute-lft-out 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y around 0 61.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos y \cdot \left(1.5 - t_0\right)\\ t_2 := \sqrt{5} \cdot 0.5\\ t_3 := 2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)\\ t_4 := t_2 - 0.5\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_3}{1 + \left(\cos x \cdot t_4 + \cos y \cdot \left(1.5 - t_2\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(t_1 + \left(1 + t_4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (* (cos y) (- 1.5 t_0)))
        (t_2 (* (sqrt 5.0) 0.5))
        (t_3
         (+
          2.0
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x)))))))
        (t_4 (- t_2 0.5)))
   (if (<= x -1.85e-6)
     (*
      0.3333333333333333
      (/ t_3 (+ 1.0 (+ (* (cos x) t_4) (* (cos y) (- 1.5 t_2))))))
     (if (<= x 0.017)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (* -0.0625 (sin y)))
          (* (- (sin y) (/ (sin x) 16.0)) (- 1.0 (cos y)))))
        (* 3.0 (+ t_1 (+ 1.0 t_4))))
       (/ t_3 (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = cos(y) * (1.5 - t_0);
	double t_2 = sqrt(5.0) * 0.5;
	double t_3 = 2.0 + (pow(sin(x), 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))));
	double t_4 = t_2 - 0.5;
	double tmp;
	if (x <= -1.85e-6) {
		tmp = 0.3333333333333333 * (t_3 / (1.0 + ((cos(x) * t_4) + (cos(y) * (1.5 - t_2)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * (t_1 + (1.0 + t_4)));
	} else {
		tmp = t_3 / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = cos(y) * (1.5d0 - t_0)
    t_2 = sqrt(5.0d0) * 0.5d0
    t_3 = 2.0d0 + ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))
    t_4 = t_2 - 0.5d0
    if (x <= (-1.85d-6)) then
        tmp = 0.3333333333333333d0 * (t_3 / (1.0d0 + ((cos(x) * t_4) + (cos(y) * (1.5d0 - t_2)))))
    else if (x <= 0.017d0) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * ((-0.0625d0) * sin(y))) * ((sin(y) - (sin(x) / 16.0d0)) * (1.0d0 - cos(y))))) / (3.0d0 * (t_1 + (1.0d0 + t_4)))
    else
        tmp = t_3 / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.cos(y) * (1.5 - t_0);
	double t_2 = Math.sqrt(5.0) * 0.5;
	double t_3 = 2.0 + (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (0.0625 + (-0.0625 * Math.cos(x)))));
	double t_4 = t_2 - 0.5;
	double tmp;
	if (x <= -1.85e-6) {
		tmp = 0.3333333333333333 * (t_3 / (1.0 + ((Math.cos(x) * t_4) + (Math.cos(y) * (1.5 - t_2)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (-0.0625 * Math.sin(y))) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (1.0 - Math.cos(y))))) / (3.0 * (t_1 + (1.0 + t_4)));
	} else {
		tmp = t_3 / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.cos(y) * (1.5 - t_0)
	t_2 = math.sqrt(5.0) * 0.5
	t_3 = 2.0 + (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (0.0625 + (-0.0625 * math.cos(x)))))
	t_4 = t_2 - 0.5
	tmp = 0
	if x <= -1.85e-6:
		tmp = 0.3333333333333333 * (t_3 / (1.0 + ((math.cos(x) * t_4) + (math.cos(y) * (1.5 - t_2)))))
	elif x <= 0.017:
		tmp = (2.0 + ((math.sqrt(2.0) * (-0.0625 * math.sin(y))) * ((math.sin(y) - (math.sin(x) / 16.0)) * (1.0 - math.cos(y))))) / (3.0 * (t_1 + (1.0 + t_4)))
	else:
		tmp = t_3 / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + t_1))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(cos(y) * Float64(1.5 - t_0))
	t_2 = Float64(sqrt(5.0) * 0.5)
	t_3 = Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x))))))
	t_4 = Float64(t_2 - 0.5)
	tmp = 0.0
	if (x <= -1.85e-6)
		tmp = Float64(0.3333333333333333 * Float64(t_3 / Float64(1.0 + Float64(Float64(cos(x) * t_4) + Float64(cos(y) * Float64(1.5 - t_2))))));
	elseif (x <= 0.017)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(1.0 - cos(y))))) / Float64(3.0 * Float64(t_1 + Float64(1.0 + t_4))));
	else
		tmp = Float64(t_3 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = cos(y) * (1.5 - t_0);
	t_2 = sqrt(5.0) * 0.5;
	t_3 = 2.0 + ((sin(x) ^ 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))));
	t_4 = t_2 - 0.5;
	tmp = 0.0;
	if (x <= -1.85e-6)
		tmp = 0.3333333333333333 * (t_3 / (1.0 + ((cos(x) * t_4) + (cos(y) * (1.5 - t_2)))));
	elseif (x <= 0.017)
		tmp = (2.0 + ((sqrt(2.0) * (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y))))) / (3.0 * (t_1 + (1.0 + t_4)));
	else
		tmp = t_3 / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - 0.5), $MachinePrecision]}, If[LessEqual[x, -1.85e-6], N[(0.3333333333333333 * N[(t$95$3 / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * t$95$4), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.017], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8500000000000001e-6

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.4%

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

      3. associate-*r* 58.4%

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

      4. *-commutative 58.4%

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

      5. sub-neg 58.4%

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

      6. metadata-eval 58.4%

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

      7. distribute-rgt-in 58.4%

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

      8. metadata-eval 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in x around inf 58.5%

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

   if -1.8500000000000001e-6 < x < 0.017000000000000001

   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%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left(\sin y \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*l* 98.8%

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

      4. metadata-eval 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. *-commutative 98.8%

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

      7. distribute-lft-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.8%

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

   9. Taylor expanded in x around 0 98.8%

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

   if 0.017000000000000001 < x

   1. Initial program 99.0%

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

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 61.9%

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

      3. associate-*r* 61.9%

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

      4. *-commutative 61.9%

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

      5. sub-neg 61.9%

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

      6. metadata-eval 61.9%

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

      7. distribute-rgt-in 61.9%

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

      8. metadata-eval 61.9%

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

   7. Simplified 61.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(-0.0625 \cdot \sin y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right)}{3 \cdot \left(\cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right) + \left(1 + \left(\sqrt{5} \cdot 0.5 - 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos y \cdot \left(1.5 - t_0\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-6} \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_0 + t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (* (cos y) (- 1.5 t_0))))
   (if (or (<= x -2.6e-6) (not (<= x 0.017)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x))))))
       (+ 1.0 (+ (* (cos x) (- t_0 0.5)) t_1))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 0.5 (+ t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = cos(y) * (1.5 - t_0);
	double tmp;
	if ((x <= -2.6e-6) || !(x <= 0.017)) {
		tmp = 0.3333333333333333 * ((2.0 + (pow(sin(x), 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + t_1)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + 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 = sqrt(5.0d0) * 0.5d0
    t_1 = cos(y) * (1.5d0 - t_0)
    if ((x <= (-2.6d-6)) .or. (.not. (x <= 0.017d0))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))) / (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + t_1)))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (0.5d0 + (t_0 + t_1)))
    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.cos(y) * (1.5 - t_0);
	double tmp;
	if ((x <= -2.6e-6) || !(x <= 0.017)) {
		tmp = 0.3333333333333333 * ((2.0 + (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (0.0625 + (-0.0625 * Math.cos(x)))))) / (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + t_1)));
	} 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 + t_1)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.cos(y) * (1.5 - t_0)
	tmp = 0
	if (x <= -2.6e-6) or not (x <= 0.017):
		tmp = 0.3333333333333333 * ((2.0 + (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (0.0625 + (-0.0625 * math.cos(x)))))) / (1.0 + ((math.cos(x) * (t_0 - 0.5)) + t_1)))
	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 + t_1)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(cos(y) * Float64(1.5 - t_0))
	tmp = 0.0
	if ((x <= -2.6e-6) || !(x <= 0.017))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x)))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + t_1))));
	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 + t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = cos(y) * (1.5 - t_0);
	tmp = 0.0;
	if ((x <= -2.6e-6) || ~((x <= 0.017)))
		tmp = 0.3333333333333333 * ((2.0 + ((sin(x) ^ 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + t_1)));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_0 + t_1)));
	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[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -2.6e-6], N[Not[LessEqual[x, 0.017]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + t$95$1), $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 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009e-6 or 0.017000000000000001 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      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 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 60.2%

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

      1. *-commutative 60.2%

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

      2. associate-*l* 60.2%

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

      3. associate-*r* 60.2%

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

      4. *-commutative 60.2%

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

      5. sub-neg 60.2%

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

      6. metadata-eval 60.2%

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

      7. distribute-rgt-in 60.2%

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

      8. metadata-eval 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in x around inf 60.2%

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

   if -2.60000000000000009e-6 < x < 0.017000000000000001

   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. +-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. fma-def 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. cos-neg 99.6%

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

      6. distribute-lft-in 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-6} \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)\\ t_2 := \sqrt{5} \cdot 0.5\\ t_3 := \cos y \cdot \left(1.5 - t_2\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + \left(\cos x \cdot \left(t_2 - 0.5\right) + t_3\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_2 + t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1
         (+
          2.0
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x)))))))
        (t_2 (* (sqrt 5.0) 0.5))
        (t_3 (* (cos y) (- 1.5 t_2))))
   (if (<= x -1.8e-6)
     (* 0.3333333333333333 (/ t_1 (+ 1.0 (+ (* (cos x) (- t_2 0.5)) t_3))))
     (if (<= x 0.017)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+ 0.5 (+ t_2 t_3))))
       (/
        t_1
        (*
         3.0
         (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = 2.0 + (pow(sin(x), 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))));
	double t_2 = sqrt(5.0) * 0.5;
	double t_3 = cos(y) * (1.5 - t_2);
	double tmp;
	if (x <= -1.8e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((cos(x) * (t_2 - 0.5)) + t_3)));
	} else if (x <= 0.017) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_2 + t_3)));
	} else {
		tmp = t_1 / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = 2.0d0 + ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))
    t_2 = sqrt(5.0d0) * 0.5d0
    t_3 = cos(y) * (1.5d0 - t_2)
    if (x <= (-1.8d-6)) then
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + ((cos(x) * (t_2 - 0.5d0)) + t_3)))
    else if (x <= 0.017d0) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (0.5d0 + (t_2 + t_3)))
    else
        tmp = t_1 / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (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) / 2.0;
	double t_1 = 2.0 + (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (0.0625 + (-0.0625 * Math.cos(x)))));
	double t_2 = Math.sqrt(5.0) * 0.5;
	double t_3 = Math.cos(y) * (1.5 - t_2);
	double tmp;
	if (x <= -1.8e-6) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((Math.cos(x) * (t_2 - 0.5)) + t_3)));
	} else if (x <= 0.017) {
		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_2 + t_3)));
	} else {
		tmp = t_1 / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = 2.0 + (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (0.0625 + (-0.0625 * math.cos(x)))))
	t_2 = math.sqrt(5.0) * 0.5
	t_3 = math.cos(y) * (1.5 - t_2)
	tmp = 0
	if x <= -1.8e-6:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((math.cos(x) * (t_2 - 0.5)) + t_3)))
	elif x <= 0.017:
		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_2 + t_3)))
	else:
		tmp = t_1 / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x))))))
	t_2 = Float64(sqrt(5.0) * 0.5)
	t_3 = Float64(cos(y) * Float64(1.5 - t_2))
	tmp = 0.0
	if (x <= -1.8e-6)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + t_3))));
	elseif (x <= 0.017)
		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_2 + t_3))));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = 2.0 + ((sin(x) ^ 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))));
	t_2 = sqrt(5.0) * 0.5;
	t_3 = cos(y) * (1.5 - t_2);
	tmp = 0.0;
	if (x <= -1.8e-6)
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((cos(x) * (t_2 - 0.5)) + t_3)));
	elseif (x <= 0.017)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (t_2 + t_3)));
	else
		tmp = t_1 / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (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] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e-6], N[(0.3333333333333333 * N[(t$95$1 / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.017], 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$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999992e-6

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*l* 58.4%

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

      3. associate-*r* 58.4%

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

      4. *-commutative 58.4%

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

      5. sub-neg 58.4%

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

      6. metadata-eval 58.4%

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

      7. distribute-rgt-in 58.4%

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

      8. metadata-eval 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in x around inf 58.5%

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

   if -1.79999999999999992e-6 < x < 0.017000000000000001

   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. +-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. fma-def 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. cos-neg 99.6%

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

      6. distribute-lft-in 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.6%

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

   if 0.017000000000000001 < x

   1. Initial program 99.0%

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

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 61.9%

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

      3. associate-*r* 61.9%

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

      4. *-commutative 61.9%

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

      5. sub-neg 61.9%

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

      6. metadata-eval 61.9%

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

      7. distribute-rgt-in 61.9%

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

      8. metadata-eval 61.9%

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

   7. Simplified 61.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)\\ t_2 := \sqrt{5} \cdot 0.5\\ t_3 := 3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\\ \mathbf{if}\;x \leq -0.00048:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{1 + \left(\cos x \cdot \left(t_2 - 0.5\right) + \cos y \cdot \left(1.5 - t_2\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1
         (+
          2.0
          (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ 0.0625 (* -0.0625 (cos x)))))))
        (t_2 (* (sqrt 5.0) 0.5))
        (t_3
         (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0))))))
   (if (<= x -0.00048)
     (*
      0.3333333333333333
      (/ t_1 (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))
     (if (<= x 0.017)
       (/
        (+
         2.0
         (* (pow (sin y) 2.0) (* (sqrt 2.0) (* -0.0625 (- 1.0 (cos y))))))
        t_3)
       (/ t_1 t_3)))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = 2.0 + (pow(sin(x), 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))));
	double t_2 = sqrt(5.0) * 0.5;
	double t_3 = 3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0)));
	double tmp;
	if (x <= -0.00048) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + (pow(sin(y), 2.0) * (sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))))) / t_3;
	} else {
		tmp = t_1 / t_3;
	}
	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 = sqrt(5.0d0) / 2.0d0
    t_1 = 2.0d0 + ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (0.0625d0 + ((-0.0625d0) * cos(x)))))
    t_2 = sqrt(5.0d0) * 0.5d0
    t_3 = 3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0)))
    if (x <= (-0.00048d0)) then
        tmp = 0.3333333333333333d0 * (t_1 / (1.0d0 + ((cos(x) * (t_2 - 0.5d0)) + (cos(y) * (1.5d0 - t_2)))))
    else if (x <= 0.017d0) then
        tmp = (2.0d0 + ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * ((-0.0625d0) * (1.0d0 - cos(y)))))) / t_3
    else
        tmp = t_1 / t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = 2.0 + (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (0.0625 + (-0.0625 * Math.cos(x)))));
	double t_2 = Math.sqrt(5.0) * 0.5;
	double t_3 = 3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0)));
	double tmp;
	if (x <= -0.00048) {
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((Math.cos(x) * (t_2 - 0.5)) + (Math.cos(y) * (1.5 - t_2)))));
	} else if (x <= 0.017) {
		tmp = (2.0 + (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (-0.0625 * (1.0 - Math.cos(y)))))) / t_3;
	} else {
		tmp = t_1 / t_3;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = 2.0 + (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (0.0625 + (-0.0625 * math.cos(x)))))
	t_2 = math.sqrt(5.0) * 0.5
	t_3 = 3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0)))
	tmp = 0
	if x <= -0.00048:
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((math.cos(x) * (t_2 - 0.5)) + (math.cos(y) * (1.5 - t_2)))))
	elif x <= 0.017:
		tmp = (2.0 + (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (-0.0625 * (1.0 - math.cos(y)))))) / t_3
	else:
		tmp = t_1 / t_3
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(0.0625 + Float64(-0.0625 * cos(x))))))
	t_2 = Float64(sqrt(5.0) * 0.5)
	t_3 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0))))
	tmp = 0.0
	if (x <= -0.00048)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	elseif (x <= 0.017)
		tmp = Float64(Float64(2.0 + Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(-0.0625 * Float64(1.0 - cos(y)))))) / t_3);
	else
		tmp = Float64(t_1 / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = 2.0 + ((sin(x) ^ 2.0) * (sqrt(2.0) * (0.0625 + (-0.0625 * cos(x)))));
	t_2 = sqrt(5.0) * 0.5;
	t_3 = 3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0)));
	tmp = 0.0;
	if (x <= -0.00048)
		tmp = 0.3333333333333333 * (t_1 / (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	elseif (x <= 0.017)
		tmp = (2.0 + ((sin(y) ^ 2.0) * (sqrt(2.0) * (-0.0625 * (1.0 - cos(y)))))) / t_3;
	else
		tmp = t_1 / t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.0625 + N[(-0.0625 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00048], N[(0.3333333333333333 * N[(t$95$1 / 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, 0.017], N[(N[(2.0 + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(t$95$1 / t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.80000000000000012e-4

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. distribute-lft-in 99.1%

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

      3. cos-neg 99.1%

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

      4. distribute-lft-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. associate-*l* 57.7%

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

      3. associate-*r* 57.7%

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

      4. *-commutative 57.7%

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

      5. sub-neg 57.7%

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

      6. metadata-eval 57.7%

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

      7. distribute-rgt-in 57.7%

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

      8. metadata-eval 57.7%

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

   7. Simplified 57.7%

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

   8. Taylor expanded in x around inf 57.8%

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

   if -4.80000000000000012e-4 < x < 0.017000000000000001

   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%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-*l* 98.8%

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

      3. associate-*r* 98.8%

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

   7. Simplified 98.8%

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

   if 0.017000000000000001 < x

   1. Initial program 99.0%

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

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 61.9%

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

      3. associate-*r* 61.9%

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

      4. *-commutative 61.9%

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

      5. sub-neg 61.9%

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

      6. metadata-eval 61.9%

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

      7. distribute-rgt-in 61.9%

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

      8. metadata-eval 61.9%

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

   7. Simplified 61.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00048:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{elif}\;x \leq 0.017:\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(0.0625 + -0.0625 \cdot \cos x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \sqrt{2} \cdot \left(\cos x + -1\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t_1 \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \mathbf{elif}\;x \leq 13:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot t_1\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (* (sqrt 2.0) (+ (cos x) -1.0))))
   (if (<= x -1.3e-5)
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* t_1 (- 0.5 (/ (cos (* 2.0 x)) 2.0)))))
       (+ (* (cos x) (+ -0.5 (sqrt 1.25))) (- 2.5 (sqrt 1.25)))))
     (if (<= x 13.0)
       (*
        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))))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) t_1)))
         (+
          1.0
          (+
           (* (* (cos x) (+ (sqrt 5.0) -1.0)) 0.5)
           (* 2.0 (/ 1.0 (+ 3.0 (sqrt 5.0))))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = sqrt(2.0) * (cos(x) + -1.0);
	double tmp;
	if (x <= -1.3e-5) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((cos(x) * (-0.5 + sqrt(1.25))) + (2.5 - sqrt(1.25))));
	} else if (x <= 13.0) {
		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)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * t_1))) / (1.0 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (1.0 / (3.0 + sqrt(5.0)))))));
	}
	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 = sqrt(5.0d0) * 0.5d0
    t_1 = sqrt(2.0d0) * (cos(x) + (-1.0d0))
    if (x <= (-1.3d-5)) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (t_1 * (0.5d0 - (cos((2.0d0 * x)) / 2.0d0))))) / ((cos(x) * ((-0.5d0) + sqrt(1.25d0))) + (2.5d0 - sqrt(1.25d0))))
    else if (x <= 13.0d0) then
        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)))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * t_1))) / (1.0d0 + (((cos(x) * (sqrt(5.0d0) + (-1.0d0))) * 0.5d0) + (2.0d0 * (1.0d0 / (3.0d0 + sqrt(5.0d0)))))))
    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.sqrt(2.0) * (Math.cos(x) + -1.0);
	double tmp;
	if (x <= -1.3e-5) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (0.5 - (Math.cos((2.0 * x)) / 2.0))))) / ((Math.cos(x) * (-0.5 + Math.sqrt(1.25))) + (2.5 - Math.sqrt(1.25))));
	} else if (x <= 13.0) {
		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)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * t_1))) / (1.0 + (((Math.cos(x) * (Math.sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (1.0 / (3.0 + Math.sqrt(5.0)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.sqrt(2.0) * (math.cos(x) + -1.0)
	tmp = 0
	if x <= -1.3e-5:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (0.5 - (math.cos((2.0 * x)) / 2.0))))) / ((math.cos(x) * (-0.5 + math.sqrt(1.25))) + (2.5 - math.sqrt(1.25))))
	elif x <= 13.0:
		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)))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * t_1))) / (1.0 + (((math.cos(x) * (math.sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (1.0 / (3.0 + math.sqrt(5.0)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(sqrt(2.0) * Float64(cos(x) + -1.0))
	tmp = 0.0
	if (x <= -1.3e-5)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(0.5 - Float64(cos(Float64(2.0 * x)) / 2.0))))) / Float64(Float64(cos(x) * Float64(-0.5 + sqrt(1.25))) + Float64(2.5 - sqrt(1.25)))));
	elseif (x <= 13.0)
		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))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * t_1))) / Float64(1.0 + Float64(Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) * 0.5) + Float64(2.0 * Float64(1.0 / Float64(3.0 + sqrt(5.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = sqrt(2.0) * (cos(x) + -1.0);
	tmp = 0.0;
	if (x <= -1.3e-5)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((cos(x) * (-0.5 + sqrt(1.25))) + (2.5 - sqrt(1.25))));
	elseif (x <= 13.0)
		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)))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * t_1))) / (1.0 + (((cos(x) * (sqrt(5.0) + -1.0)) * 0.5) + (2.0 * (1.0 / (3.0 + sqrt(5.0)))))));
	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[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-5], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.5 - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 13.0], 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], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(2.0 * N[(1.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.29999999999999992e-5

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. fma-def 99.1%

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

      4. distribute-lft-in 99.1%

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

      5. cos-neg 99.1%

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

      6. distribute-lft-in 99.1%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around 0 56.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 56.4%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right) \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}}} \]

      2. pow3 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right)}^{3}}} \]

      3. cancel-sign-sub-inv 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + \left(-0.5\right) \cdot \sqrt{5}}}\right)}^{3}} \]

      4. +-commutative 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      5. fma-def 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\cos x, 0.5 \cdot \sqrt{5} - 0.5, 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      6. *-commutative 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} \cdot 0.5} - 0.5, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      7. fma-neg 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)}, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      8. metadata-eval 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, \color{blue}{-0.5}\right), 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      9. metadata-eval 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + \color{blue}{-0.5} \cdot \sqrt{5}}\right)}^{3}} \]

   7. Applied egg-rr 56.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

      2. sin-mult 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   9. Applied egg-rr 56.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. div-sub 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       2. +-inverses 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       3. cos-0 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       4. metadata-eval 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       5. count-2 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       6. *-commutative 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   11. Simplified 56.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   13. Applied egg-rr 57.0%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}} \]

   if -1.29999999999999992e-5 < x < 13

   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. +-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. fma-def 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. cos-neg 99.6%

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

      6. distribute-lft-in 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.0%

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

   if 13 < x

   1. Initial program 99.0%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

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

      3. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y + -0.0625 \cdot \sin x\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

      8. metadata-eval 99.1%

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in y around 0 61.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \mathbf{elif}\;x \leq 13:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot 0.5 + 2 \cdot \frac{1}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := 2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)\\ t_2 := -0.5 + \sqrt{1.25}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{\cos x \cdot t_2 + \left(2.5 - \sqrt{1.25}\right)}\\ \mathbf{elif}\;x \leq 13:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_1}{\mathsf{fma}\left(\cos x, t_2, 2.5\right) - \sqrt{1.25}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1
         (+
          2.0
          (*
           -0.0625
           (*
            (* (sqrt 2.0) (+ (cos x) -1.0))
            (- 0.5 (/ (cos (* 2.0 x)) 2.0))))))
        (t_2 (+ -0.5 (sqrt 1.25))))
   (if (<= x -2.5e-5)
     (* 0.3333333333333333 (/ t_1 (+ (* (cos x) t_2) (- 2.5 (sqrt 1.25)))))
     (if (<= x 13.0)
       (*
        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))))))
       (* 0.3333333333333333 (/ t_1 (- (fma (cos x) t_2 2.5) (sqrt 1.25))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = 2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))));
	double t_2 = -0.5 + sqrt(1.25);
	double tmp;
	if (x <= -2.5e-5) {
		tmp = 0.3333333333333333 * (t_1 / ((cos(x) * t_2) + (2.5 - sqrt(1.25))));
	} else if (x <= 13.0) {
		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)))));
	} else {
		tmp = 0.3333333333333333 * (t_1 / (fma(cos(x), t_2, 2.5) - sqrt(1.25)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * Float64(cos(x) + -1.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * x)) / 2.0)))))
	t_2 = Float64(-0.5 + sqrt(1.25))
	tmp = 0.0
	if (x <= -2.5e-5)
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(Float64(cos(x) * t_2) + Float64(2.5 - sqrt(1.25)))));
	elseif (x <= 13.0)
		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))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(t_1 / Float64(fma(cos(x), t_2, 2.5) - sqrt(1.25))));
	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[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-5], N[(0.3333333333333333 * N[(t$95$1 / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(2.5 - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 13.0], 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], N[(0.3333333333333333 * N[(t$95$1 / N[(N[(N[Cos[x], $MachinePrecision] * t$95$2 + 2.5), $MachinePrecision] - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000012e-5

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. fma-def 99.1%

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

      4. distribute-lft-in 99.1%

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

      5. cos-neg 99.1%

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

      6. distribute-lft-in 99.1%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around 0 56.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 56.4%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right) \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}}} \]

      2. pow3 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right)}^{3}}} \]

      3. cancel-sign-sub-inv 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + \left(-0.5\right) \cdot \sqrt{5}}}\right)}^{3}} \]

      4. +-commutative 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      5. fma-def 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\cos x, 0.5 \cdot \sqrt{5} - 0.5, 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      6. *-commutative 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} \cdot 0.5} - 0.5, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      7. fma-neg 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)}, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      8. metadata-eval 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, \color{blue}{-0.5}\right), 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      9. metadata-eval 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + \color{blue}{-0.5} \cdot \sqrt{5}}\right)}^{3}} \]

   7. Applied egg-rr 56.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

      2. sin-mult 56.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   9. Applied egg-rr 56.3%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. div-sub 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       2. +-inverses 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       3. cos-0 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       4. metadata-eval 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       5. count-2 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       6. *-commutative 56.3%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   11. Simplified 56.3%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   13. Applied egg-rr 57.0%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}} \]

   if -2.50000000000000012e-5 < x < 13

   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. +-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. fma-def 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. cos-neg 99.6%

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

      6. distribute-lft-in 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.0%

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

   if 13 < x

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. associate-*l* 98.9%

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

      3. fma-def 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. cos-neg 99.0%

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

      6. distribute-lft-in 99.0%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 61.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 60.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right) \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}}} \]

      2. pow3 60.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right)}^{3}}} \]

      3. cancel-sign-sub-inv 60.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + \left(-0.5\right) \cdot \sqrt{5}}}\right)}^{3}} \]

      4. +-commutative 60.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      5. fma-def 60.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\cos x, 0.5 \cdot \sqrt{5} - 0.5, 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      6. *-commutative 60.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} \cdot 0.5} - 0.5, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      7. fma-neg 60.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)}, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      8. metadata-eval 60.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, \color{blue}{-0.5}\right), 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

      9. metadata-eval 60.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + \color{blue}{-0.5} \cdot \sqrt{5}}\right)}^{3}} \]

   7. Applied egg-rr 60.5%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 60.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

      2. sin-mult 60.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   9. Applied egg-rr 60.5%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. div-sub 60.5%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       2. +-inverses 60.5%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       3. cos-0 60.5%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       4. metadata-eval 60.5%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       5. count-2 60.5%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

       6. *-commutative 60.5%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   11. Simplified 60.5%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   13. Applied egg-rr 61.1%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, 2.5\right) - \sqrt{1.25}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)}\\ \mathbf{elif}\;x \leq 13:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\mathsf{fma}\left(\cos x, -0.5 + \sqrt{1.25}, 2.5\right) - \sqrt{1.25}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 59.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\mathsf{fma}\left(\cos x, -0.5 + \sqrt{1.25}, 2.5\right) - \sqrt{1.25}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+
    2.0
    (*
     -0.0625
     (* (* (sqrt 2.0) (+ (cos x) -1.0)) (- 0.5 (/ (cos (* 2.0 x)) 2.0)))))
   (- (fma (cos x) (+ -0.5 (sqrt 1.25)) 2.5) (sqrt 1.25)))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))))) / (fma(cos(x), (-0.5 + sqrt(1.25)), 2.5) - sqrt(1.25)));
}

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * Float64(cos(x) + -1.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * x)) / 2.0))))) / Float64(fma(cos(x), Float64(-0.5 + sqrt(1.25)), 2.5) - sqrt(1.25))))
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision] + 2.5), $MachinePrecision] - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. distribute-lft-in 99.3%

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

   5. cos-neg 99.3%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 60.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
6. Step-by-step derivation

   1. add-cube-cbrt 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right) \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}}} \]

   2. pow3 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right)}^{3}}} \]

   3. cancel-sign-sub-inv 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + \left(-0.5\right) \cdot \sqrt{5}}}\right)}^{3}} \]

   4. +-commutative 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   5. fma-def 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\cos x, 0.5 \cdot \sqrt{5} - 0.5, 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   6. *-commutative 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} \cdot 0.5} - 0.5, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   7. fma-neg 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)}, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   8. metadata-eval 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, \color{blue}{-0.5}\right), 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   9. metadata-eval 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + \color{blue}{-0.5} \cdot \sqrt{5}}\right)}^{3}} \]

7. Applied egg-rr 59.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}}} \]
8. Step-by-step derivation

   1. unpow2 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   2. sin-mult 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

9. Applied egg-rr 59.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]
10. Step-by-step derivation

    1. div-sub 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    2. +-inverses 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    3. cos-0 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    4. metadata-eval 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    5. count-2 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    6. *-commutative 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

11. Simplified 59.2%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

13. Applied egg-rr 60.4%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, 2.5\right) - \sqrt{1.25}}} \]

14. Final simplification 60.4%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\mathsf{fma}\left(\cos x, -0.5 + \sqrt{1.25}, 2.5\right) - \sqrt{1.25}} \]
15. Add Preprocessing

Alternative 26: 59.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (*
  0.3333333333333333
  (/
   (+
    2.0
    (*
     -0.0625
     (* (* (sqrt 2.0) (+ (cos x) -1.0)) (- 0.5 (/ (cos (* 2.0 x)) 2.0)))))
   (+ (* (cos x) (+ -0.5 (sqrt 1.25))) (- 2.5 (sqrt 1.25))))))

C

double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((cos(x) * (-0.5 + sqrt(1.25))) + (2.5 - sqrt(1.25))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sqrt(2.0d0) * (cos(x) + (-1.0d0))) * (0.5d0 - (cos((2.0d0 * x)) / 2.0d0))))) / ((cos(x) * ((-0.5d0) + sqrt(1.25d0))) + (2.5d0 - sqrt(1.25d0))))
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((Math.sqrt(2.0) * (Math.cos(x) + -1.0)) * (0.5 - (Math.cos((2.0 * x)) / 2.0))))) / ((Math.cos(x) * (-0.5 + Math.sqrt(1.25))) + (2.5 - Math.sqrt(1.25))));
}

Python

def code(x, y):
	return 0.3333333333333333 * ((2.0 + (-0.0625 * ((math.sqrt(2.0) * (math.cos(x) + -1.0)) * (0.5 - (math.cos((2.0 * x)) / 2.0))))) / ((math.cos(x) * (-0.5 + math.sqrt(1.25))) + (2.5 - math.sqrt(1.25))))

Julia

function code(x, y)
	return Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * Float64(cos(x) + -1.0)) * Float64(0.5 - Float64(cos(Float64(2.0 * x)) / 2.0))))) / Float64(Float64(cos(x) * Float64(-0.5 + sqrt(1.25))) + Float64(2.5 - sqrt(1.25)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (cos(x) + -1.0)) * (0.5 - (cos((2.0 * x)) / 2.0))))) / ((cos(x) * (-0.5 + sqrt(1.25))) + (2.5 - sqrt(1.25))));
end

Wolfram

code[x_, y_] := N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.5 - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. distribute-lft-in 99.3%

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

   5. cos-neg 99.3%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 60.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]
6. Step-by-step derivation

   1. add-cube-cbrt 59.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}} \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right) \cdot \sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}}} \]

   2. pow3 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}\right)}^{3}}} \]

   3. cancel-sign-sub-inv 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) + \left(-0.5\right) \cdot \sqrt{5}}}\right)}^{3}} \]

   4. +-commutative 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   5. fma-def 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\cos x, 0.5 \cdot \sqrt{5} - 0.5, 2.5\right)} + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   6. *-commutative 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\sqrt{5} \cdot 0.5} - 0.5, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   7. fma-neg 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \color{blue}{\mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right)}, 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   8. metadata-eval 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, \color{blue}{-0.5}\right), 2.5\right) + \left(-0.5\right) \cdot \sqrt{5}}\right)}^{3}} \]

   9. metadata-eval 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + \color{blue}{-0.5} \cdot \sqrt{5}}\right)}^{3}} \]

7. Applied egg-rr 59.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}}} \]
8. Step-by-step derivation

   1. unpow2 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

   2. sin-mult 59.2%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

9. Applied egg-rr 59.2%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]
10. Step-by-step derivation

    1. div-sub 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    2. +-inverses 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    3. cos-0 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    4. metadata-eval 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    5. count-2 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

    6. *-commutative 59.2%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

11. Simplified 59.2%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\color{blue}{\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right)} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{{\left(\sqrt[3]{\mathsf{fma}\left(\cos x, \mathsf{fma}\left(\sqrt{5}, 0.5, -0.5\right), 2.5\right) + -0.5 \cdot \sqrt{5}}\right)}^{3}} \]

13. Applied egg-rr 60.4%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(0.5 - \frac{\cos \left(x \cdot 2\right)}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(2.5 - \sqrt{1.25}\right)}} \]

14. Final simplification 60.4%

    \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(\cos x + -1\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot x\right)}{2}\right)\right)}{\cos x \cdot \left(-0.5 + \sqrt{1.25}\right) + \left(2.5 - \sqrt{1.25}\right)} \]
15. Add Preprocessing

Alternative 27: 42.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{0.6666666666666666}{0.5 + \left(t_0 + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (/ 0.6666666666666666 (+ 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;
	return 0.6666666666666666 / (0.5 + (t_0 + (cos(y) * (1.5 - t_0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) * 0.5d0
    code = 0.6666666666666666d0 / (0.5d0 + (t_0 + (cos(y) * (1.5d0 - t_0))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	return 0.6666666666666666 / (0.5 + (t_0 + (Math.cos(y) * (1.5 - t_0))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	return 0.6666666666666666 / (0.5 + (t_0 + (math.cos(y) * (1.5 - t_0))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	return Float64(0.6666666666666666 / Float64(0.5 + Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0)))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.6666666666666666 / (0.5 + (t_0 + (cos(y) * (1.5 - t_0))));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(0.6666666666666666 / N[(0.5 + N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-lft-in 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around 0 62.6%

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

   1. *-commutative 62.6%

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

   2. associate-*l* 62.6%

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

   3. associate-*r* 62.6%

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

   4. *-commutative 62.6%

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

   5. sub-neg 62.6%

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

   6. metadata-eval 62.6%

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

   7. distribute-rgt-in 62.6%

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

   8. metadata-eval 62.6%

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

7. Simplified 62.6%

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

8. Taylor expanded in x around 0 42.4%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{0.5 + \left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right)}} \]

9. Final simplification 42.4%

   \[\leadsto \frac{0.6666666666666666}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)} \]
10. Add Preprocessing

Alternative 28: 40.6% 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 99.3%

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

   1. +-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. distribute-lft-in 99.3%

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

   5. cos-neg 99.3%

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

   6. distribute-lft-in 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 60.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\left(2.5 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right) - 0.5 \cdot \sqrt{5}}} \]

6. Taylor expanded in x around 0 40.7%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right)}{\color{blue}{2}} \]

7. Taylor expanded in x around 0 31.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left(-0.5 \cdot \left({x}^{4} \cdot \sqrt{2}\right)\right)}}{2} \]
8. Step-by-step derivation

   1. *-commutative 31.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left(\left({x}^{4} \cdot \sqrt{2}\right) \cdot -0.5\right)}}{2} \]

   2. associate-*l* 31.6%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left({x}^{4} \cdot \left(\sqrt{2} \cdot -0.5\right)\right)}}{2} \]

9. Simplified 31.6%

   \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \color{blue}{\left({x}^{4} \cdot \left(\sqrt{2} \cdot -0.5\right)\right)}}{2} \]

10. Taylor expanded in x around 0 40.6%

    \[\leadsto \color{blue}{0.3333333333333333} \]

11. Final simplification 40.6%

    \[\leadsto 0.3333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))))