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

Percentage Accurate: 99.2% → 99.3%

Time: 51.5s

Alternatives: 26

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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

   4. sub-neg 99.4%

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

   5. metadata-eval 99.4%

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

   6. distribute-lft-in 99.4%

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

   7. metadata-eval 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)\\ t_2 := \cos x - \cos y\\ \mathbf{if}\;x \leq -0.0135:\\ \;\;\;\;\frac{2 + \left(t\_2 \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{1.5 \cdot \left(\cos y \cdot t\_0\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ t_0 2.0)))))
        (t_2 (- (cos x) (cos y))))
   (if (<= x -0.0135)
     (/
      (+
       2.0
       (* (* t_2 (+ (sin y) (* (sin x) -0.0625))) (* (sqrt 2.0) (sin x))))
      (+
       (* 1.5 (* (cos y) t_0))
       (+ 3.0 (* (cos x) (* 3.0 (+ (/ (sqrt 5.0) 2.0) -0.5))))))
     (if (<= x 0.022)
       (/
        (+
         2.0
         (*
          (*
           (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
           (- (sin y) (/ (sin x) 16.0)))
          (- 1.0 (cos y))))
        t_1)
       (/
        (+
         2.0
         (* t_2 (* (sin x) (* (sqrt 2.0) (- (sin y) (* (sin x) 0.0625))))))
        t_1)))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0)));
	double t_2 = cos(x) - cos(y);
	double tmp;
	if (x <= -0.0135) {
		tmp = (2.0 + ((t_2 * (sin(y) + (sin(x) * -0.0625))) * (sqrt(2.0) * sin(x)))) / ((1.5 * (cos(y) * t_0)) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))));
	} else if (x <= 0.022) {
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * (t_0 / 2.0d0)))
    t_2 = cos(x) - cos(y)
    if (x <= (-0.0135d0)) then
        tmp = (2.0d0 + ((t_2 * (sin(y) + (sin(x) * (-0.0625d0)))) * (sqrt(2.0d0) * sin(x)))) / ((1.5d0 * (cos(y) * t_0)) + (3.0d0 + (cos(x) * (3.0d0 * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0))))))
    else if (x <= 0.022d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (sin(y) - (sin(x) / 16.0d0))) * (1.0d0 - cos(y)))) / t_1
    else
        tmp = (2.0d0 + (t_2 * (sin(x) * (sqrt(2.0d0) * (sin(y) - (sin(x) * 0.0625d0)))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * (t_0 / 2.0)));
	double t_2 = Math.cos(x) - Math.cos(y);
	double tmp;
	if (x <= -0.0135) {
		tmp = (2.0 + ((t_2 * (Math.sin(y) + (Math.sin(x) * -0.0625))) * (Math.sqrt(2.0) * Math.sin(x)))) / ((1.5 * (Math.cos(y) * t_0)) + (3.0 + (Math.cos(x) * (3.0 * ((Math.sqrt(5.0) / 2.0) + -0.5)))));
	} else if (x <= 0.022) {
		tmp = (2.0 + (((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (1.0 - Math.cos(y)))) / t_1;
	} else {
		tmp = (2.0 + (t_2 * (Math.sin(x) * (Math.sqrt(2.0) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * (t_0 / 2.0)))
	t_2 = math.cos(x) - math.cos(y)
	tmp = 0
	if x <= -0.0135:
		tmp = (2.0 + ((t_2 * (math.sin(y) + (math.sin(x) * -0.0625))) * (math.sqrt(2.0) * math.sin(x)))) / ((1.5 * (math.cos(y) * t_0)) + (3.0 + (math.cos(x) * (3.0 * ((math.sqrt(5.0) / 2.0) + -0.5)))))
	elif x <= 0.022:
		tmp = (2.0 + (((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (math.sin(y) - (math.sin(x) / 16.0))) * (1.0 - math.cos(y)))) / t_1
	else:
		tmp = (2.0 + (t_2 * (math.sin(x) * (math.sqrt(2.0) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0))))
	t_2 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (x <= -0.0135)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * Float64(sin(y) + Float64(sin(x) * -0.0625))) * Float64(sqrt(2.0) * sin(x)))) / Float64(Float64(1.5 * Float64(cos(y) * t_0)) + Float64(3.0 + Float64(cos(x) * Float64(3.0 * Float64(Float64(sqrt(5.0) / 2.0) + -0.5))))));
	elseif (x <= 0.022)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(1.0 - cos(y)))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(sin(x) * Float64(sqrt(2.0) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0)));
	t_2 = cos(x) - cos(y);
	tmp = 0.0;
	if (x <= -0.0135)
		tmp = (2.0 + ((t_2 * (sin(y) + (sin(x) * -0.0625))) * (sqrt(2.0) * sin(x)))) / ((1.5 * (cos(y) * t_0)) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))));
	elseif (x <= 0.022)
		tmp = (2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (1.0 - cos(y)))) / t_1;
	else
		tmp = (2.0 + (t_2 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0135], N[(N[(2.0 + N[(N[(t$95$2 * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.022], 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[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0134999999999999998

   1. Initial program 98.6%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. metadata-eval 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*r* 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 63.6%

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

      1. *-commutative 63.6%

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

   10. Simplified 63.6%

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

   if -0.0134999999999999998 < x < 0.021999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   if 0.021999999999999999 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 60.5%

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

   4. Taylor expanded in x around inf 60.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0135:\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-5} \lor \neg \left(x \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(t\_1 \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{\cos y \cdot \left(t\_0 \cdot 1.5\right) + \left(3 + \left(\sqrt{5} \cdot 1.5 + -1.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (- (cos x) (cos y))))
   (if (or (<= x -4.6e-5) (not (<= x 5.3e-5)))
     (/
      (+ 2.0 (* t_1 (* (sin x) (* (sqrt 2.0) (- (sin y) (* (sin x) 0.0625))))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ t_0 2.0)))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (+ (sin x) (* (sin y) -0.0625))
         (* t_1 (+ (sin y) (* (sin x) -0.0625))))))
      (+ (* (cos y) (* t_0 1.5)) (+ 3.0 (+ (* (sqrt 5.0) 1.5) -1.5)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double tmp;
	if ((x <= -4.6e-5) || !(x <= 5.3e-5)) {
		tmp = (2.0 + (t_1 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * (t_1 * (sin(y) + (sin(x) * -0.0625)))))) / ((cos(y) * (t_0 * 1.5)) + (3.0 + ((sqrt(5.0) * 1.5) + -1.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = cos(x) - cos(y)
    if ((x <= (-4.6d-5)) .or. (.not. (x <= 5.3d-5))) then
        tmp = (2.0d0 + (t_1 * (sin(x) * (sqrt(2.0d0) * (sin(y) - (sin(x) * 0.0625d0)))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * (t_0 / 2.0d0))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + (sin(y) * (-0.0625d0))) * (t_1 * (sin(y) + (sin(x) * (-0.0625d0))))))) / ((cos(y) * (t_0 * 1.5d0)) + (3.0d0 + ((sqrt(5.0d0) * 1.5d0) + (-1.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.cos(x) - Math.cos(y);
	double tmp;
	if ((x <= -4.6e-5) || !(x <= 5.3e-5)) {
		tmp = (2.0 + (t_1 * (Math.sin(x) * (Math.sqrt(2.0) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * (t_0 / 2.0))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (Math.sin(y) * -0.0625)) * (t_1 * (Math.sin(y) + (Math.sin(x) * -0.0625)))))) / ((Math.cos(y) * (t_0 * 1.5)) + (3.0 + ((Math.sqrt(5.0) * 1.5) + -1.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.cos(x) - math.cos(y)
	tmp = 0
	if (x <= -4.6e-5) or not (x <= 5.3e-5):
		tmp = (2.0 + (t_1 * (math.sin(x) * (math.sqrt(2.0) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * (t_0 / 2.0))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (math.sin(y) * -0.0625)) * (t_1 * (math.sin(y) + (math.sin(x) * -0.0625)))))) / ((math.cos(y) * (t_0 * 1.5)) + (3.0 + ((math.sqrt(5.0) * 1.5) + -1.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if ((x <= -4.6e-5) || !(x <= 5.3e-5))
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(sin(x) * Float64(sqrt(2.0) * Float64(sin(y) - Float64(sin(x) * 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(t_0 / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(t_1 * Float64(sin(y) + Float64(sin(x) * -0.0625)))))) / Float64(Float64(cos(y) * Float64(t_0 * 1.5)) + Float64(3.0 + Float64(Float64(sqrt(5.0) * 1.5) + -1.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = cos(x) - cos(y);
	tmp = 0.0;
	if ((x <= -4.6e-5) || ~((x <= 5.3e-5)))
		tmp = (2.0 + (t_1 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * (t_1 * (sin(y) + (sin(x) * -0.0625)))))) / ((cos(y) * (t_0 * 1.5)) + (3.0 + ((sqrt(5.0) * 1.5) + -1.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.6e-5], N[Not[LessEqual[x, 5.3e-5]], $MachinePrecision]], N[(N[(2.0 + N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $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[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 * 1.5), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6e-5 or 5.3000000000000001e-5 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 62.0%

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

   4. Taylor expanded in x around inf 62.0%

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

   if -4.6e-5 < x < 5.3000000000000001e-5

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-5} \lor \neg \left(x \leq 5.3 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \left(3 + \left(\sqrt{5} \cdot 1.5 + -1.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \cos x - \cos y\\ t_2 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos x \cdot t\_2\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00048:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(t\_1 \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{\cos y \cdot \left(t\_0 \cdot 1.5\right) + \left(3 + \left(\sqrt{5} \cdot 1.5 + -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t\_2}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (- (cos x) (cos y)))
        (t_2 (+ (sqrt 5.0) -1.0)))
   (if (<= x -3.9e-5)
     (/
      (+ 2.0 (* t_1 (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      (*
       3.0
       (+
        1.0
        (+ (* 0.5 (* (cos x) t_2)) (* 2.0 (/ (cos y) (+ 3.0 (sqrt 5.0))))))))
     (if (<= x 0.00048)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (*
           (+ (sin x) (* (sin y) -0.0625))
           (* t_1 (+ (sin y) (* (sin x) -0.0625))))))
        (+ (* (cos y) (* t_0 1.5)) (+ 3.0 (+ (* (sqrt 5.0) 1.5) -1.5))))
       (/
        (+
         2.0
         (* t_1 (* (sin x) (* (sqrt 2.0) (- (sin y) (* (sin x) 0.0625))))))
        (*
         3.0
         (+ (+ 1.0 (* (cos x) (/ t_2 2.0))) (* (cos y) (/ t_0 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -3.9e-5) {
		tmp = (2.0 + (t_1 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((0.5 * (cos(x) * t_2)) + (2.0 * (cos(y) / (3.0 + sqrt(5.0)))))));
	} else if (x <= 0.00048) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * (t_1 * (sin(y) + (sin(x) * -0.0625)))))) / ((cos(y) * (t_0 * 1.5)) + (3.0 + ((sqrt(5.0) * 1.5) + -1.5)));
	} else {
		tmp = (2.0 + (t_1 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_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 = 3.0d0 - sqrt(5.0d0)
    t_1 = cos(x) - cos(y)
    t_2 = sqrt(5.0d0) + (-1.0d0)
    if (x <= (-3.9d-5)) then
        tmp = (2.0d0 + (t_1 * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * (1.0d0 + ((0.5d0 * (cos(x) * t_2)) + (2.0d0 * (cos(y) / (3.0d0 + sqrt(5.0d0)))))))
    else if (x <= 0.00048d0) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + (sin(y) * (-0.0625d0))) * (t_1 * (sin(y) + (sin(x) * (-0.0625d0))))))) / ((cos(y) * (t_0 * 1.5d0)) + (3.0d0 + ((sqrt(5.0d0) * 1.5d0) + (-1.5d0))))
    else
        tmp = (2.0d0 + (t_1 * (sin(x) * (sqrt(2.0d0) * (sin(y) - (sin(x) * 0.0625d0)))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_2 / 2.0d0))) + (cos(y) * (t_0 / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.cos(x) - Math.cos(y);
	double t_2 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -3.9e-5) {
		tmp = (2.0 + (t_1 * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((0.5 * (Math.cos(x) * t_2)) + (2.0 * (Math.cos(y) / (3.0 + Math.sqrt(5.0)))))));
	} else if (x <= 0.00048) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (Math.sin(y) * -0.0625)) * (t_1 * (Math.sin(y) + (Math.sin(x) * -0.0625)))))) / ((Math.cos(y) * (t_0 * 1.5)) + (3.0 + ((Math.sqrt(5.0) * 1.5) + -1.5)));
	} else {
		tmp = (2.0 + (t_1 * (Math.sin(x) * (Math.sqrt(2.0) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_2 / 2.0))) + (Math.cos(y) * (t_0 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.cos(x) - math.cos(y)
	t_2 = math.sqrt(5.0) + -1.0
	tmp = 0
	if x <= -3.9e-5:
		tmp = (2.0 + (t_1 * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 * (1.0 + ((0.5 * (math.cos(x) * t_2)) + (2.0 * (math.cos(y) / (3.0 + math.sqrt(5.0)))))))
	elif x <= 0.00048:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (math.sin(y) * -0.0625)) * (t_1 * (math.sin(y) + (math.sin(x) * -0.0625)))))) / ((math.cos(y) * (t_0 * 1.5)) + (3.0 + ((math.sqrt(5.0) * 1.5) + -1.5)))
	else:
		tmp = (2.0 + (t_1 * (math.sin(x) * (math.sqrt(2.0) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (math.cos(x) * (t_2 / 2.0))) + (math.cos(y) * (t_0 / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -3.9e-5)
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(0.5 * Float64(cos(x) * t_2)) + Float64(2.0 * Float64(cos(y) / Float64(3.0 + sqrt(5.0))))))));
	elseif (x <= 0.00048)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(t_1 * Float64(sin(y) + Float64(sin(x) * -0.0625)))))) / Float64(Float64(cos(y) * Float64(t_0 * 1.5)) + Float64(3.0 + Float64(Float64(sqrt(5.0) * 1.5) + -1.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(sin(x) * Float64(sqrt(2.0) * Float64(sin(y) - Float64(sin(x) * 0.0625)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = cos(x) - cos(y);
	t_2 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if (x <= -3.9e-5)
		tmp = (2.0 + (t_1 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + ((0.5 * (cos(x) * t_2)) + (2.0 * (cos(y) / (3.0 + sqrt(5.0)))))));
	elseif (x <= 0.00048)
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * (t_1 * (sin(y) + (sin(x) * -0.0625)))))) / ((cos(y) * (t_0 * 1.5)) + (3.0 + ((sqrt(5.0) * 1.5) + -1.5)));
	else
		tmp = (2.0 + (t_1 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * (t_2 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3.9e-5], N[(N[(2.0 + N[(t$95$1 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(0.5 * N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Cos[y], $MachinePrecision] / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00048], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 * 1.5), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $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]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8999999999999999e-5

   1. Initial program 98.6%

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

      1. flip-- 59.9%

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

      2. metadata-eval 59.9%

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

      3. pow1/2 59.9%

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

      4. pow1/2 59.9%

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

      5. pow-prod-up 60.1%

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

      6. metadata-eval 60.1%

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

      7. metadata-eval 60.1%

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

      8. metadata-eval 60.1%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in y around 0 63.5%

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

   6. Taylor expanded in x around inf 63.5%

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

   if -3.8999999999999999e-5 < x < 4.80000000000000012e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

   if 4.80000000000000012e-4 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 60.5%

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

   4. Taylor expanded in x around inf 60.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) + 2 \cdot \frac{\cos y}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00048:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \left(3 + \left(\sqrt{5} \cdot 1.5 + -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := t\_0 \cdot \left(\sin y + \sin x \cdot -0.0625\right)\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(\sqrt{2} \cdot \sin x\right)}{1.5 \cdot \left(\cos y \cdot t\_2\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0002:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot t\_1\right)}{\cos y \cdot \left(t\_2 \cdot 1.5\right) + \left(3 + \left(\sqrt{5} \cdot 1.5 + -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t\_0 \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_2}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (* t_0 (+ (sin y) (* (sin x) -0.0625))))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (<= x -2.25e-5)
     (/
      (+ 2.0 (* t_1 (* (sqrt 2.0) (sin x))))
      (+
       (* 1.5 (* (cos y) t_2))
       (+ 3.0 (* (cos x) (* 3.0 (+ (/ (sqrt 5.0) 2.0) -0.5))))))
     (if (<= x 0.0002)
       (/
        (+ 2.0 (* (sqrt 2.0) (* (+ (sin x) (* (sin y) -0.0625)) t_1)))
        (+ (* (cos y) (* t_2 1.5)) (+ 3.0 (+ (* (sqrt 5.0) 1.5) -1.5))))
       (/
        (+
         2.0
         (* t_0 (* (sin x) (* (sqrt 2.0) (- (sin y) (* (sin x) 0.0625))))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ t_2 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = t_0 * (sin(y) + (sin(x) * -0.0625));
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -2.25e-5) {
		tmp = (2.0 + (t_1 * (sqrt(2.0) * sin(x)))) / ((1.5 * (cos(y) * t_2)) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))));
	} else if (x <= 0.0002) {
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * t_1))) / ((cos(y) * (t_2 * 1.5)) + (3.0 + ((sqrt(5.0) * 1.5) + -1.5)));
	} else {
		tmp = (2.0 + (t_0 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_2 / 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 = cos(x) - cos(y)
    t_1 = t_0 * (sin(y) + (sin(x) * (-0.0625d0)))
    t_2 = 3.0d0 - sqrt(5.0d0)
    if (x <= (-2.25d-5)) then
        tmp = (2.0d0 + (t_1 * (sqrt(2.0d0) * sin(x)))) / ((1.5d0 * (cos(y) * t_2)) + (3.0d0 + (cos(x) * (3.0d0 * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0))))))
    else if (x <= 0.0002d0) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((sin(x) + (sin(y) * (-0.0625d0))) * t_1))) / ((cos(y) * (t_2 * 1.5d0)) + (3.0d0 + ((sqrt(5.0d0) * 1.5d0) + (-1.5d0))))
    else
        tmp = (2.0d0 + (t_0 * (sin(x) * (sqrt(2.0d0) * (sin(y) - (sin(x) * 0.0625d0)))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * (t_2 / 2.0d0))))
    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 = t_0 * (Math.sin(y) + (Math.sin(x) * -0.0625));
	double t_2 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if (x <= -2.25e-5) {
		tmp = (2.0 + (t_1 * (Math.sqrt(2.0) * Math.sin(x)))) / ((1.5 * (Math.cos(y) * t_2)) + (3.0 + (Math.cos(x) * (3.0 * ((Math.sqrt(5.0) / 2.0) + -0.5)))));
	} else if (x <= 0.0002) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.sin(x) + (Math.sin(y) * -0.0625)) * t_1))) / ((Math.cos(y) * (t_2 * 1.5)) + (3.0 + ((Math.sqrt(5.0) * 1.5) + -1.5)));
	} else {
		tmp = (2.0 + (t_0 * (Math.sin(x) * (Math.sqrt(2.0) * (Math.sin(y) - (Math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * (t_2 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = t_0 * (math.sin(y) + (math.sin(x) * -0.0625))
	t_2 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if x <= -2.25e-5:
		tmp = (2.0 + (t_1 * (math.sqrt(2.0) * math.sin(x)))) / ((1.5 * (math.cos(y) * t_2)) + (3.0 + (math.cos(x) * (3.0 * ((math.sqrt(5.0) / 2.0) + -0.5)))))
	elif x <= 0.0002:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.sin(x) + (math.sin(y) * -0.0625)) * t_1))) / ((math.cos(y) * (t_2 * 1.5)) + (3.0 + ((math.sqrt(5.0) * 1.5) + -1.5)))
	else:
		tmp = (2.0 + (t_0 * (math.sin(x) * (math.sqrt(2.0) * (math.sin(y) - (math.sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * (t_2 / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(t_0 * Float64(sin(y) + Float64(sin(x) * -0.0625)))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -2.25e-5)
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(sqrt(2.0) * sin(x)))) / Float64(Float64(1.5 * Float64(cos(y) * t_2)) + Float64(3.0 + Float64(cos(x) * Float64(3.0 * Float64(Float64(sqrt(5.0) / 2.0) + -0.5))))));
	elseif (x <= 0.0002)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * t_1))) / Float64(Float64(cos(y) * Float64(t_2 * 1.5)) + Float64(3.0 + Float64(Float64(sqrt(5.0) * 1.5) + -1.5))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(sin(x) * Float64(sqrt(2.0) * Float64(sin(y) - Float64(sin(x) * 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(t_2 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = t_0 * (sin(y) + (sin(x) * -0.0625));
	t_2 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if (x <= -2.25e-5)
		tmp = (2.0 + (t_1 * (sqrt(2.0) * sin(x)))) / ((1.5 * (cos(y) * t_2)) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))));
	elseif (x <= 0.0002)
		tmp = (2.0 + (sqrt(2.0) * ((sin(x) + (sin(y) * -0.0625)) * t_1))) / ((cos(y) * (t_2 * 1.5)) + (3.0 + ((sqrt(5.0) * 1.5) + -1.5)));
	else
		tmp = (2.0 + (t_0 * (sin(x) * (sqrt(2.0) * (sin(y) - (sin(x) * 0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_2 / 2.0))));
	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[(t$95$0 * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.25e-5], N[(N[(2.0 + N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0002], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 * 1.5), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 1.5), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * 0.0625), $MachinePrecision]), $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[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.25000000000000014e-5

   1. Initial program 98.6%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. metadata-eval 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*r* 98.9%

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

      5. *-commutative 98.9%

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

      6. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 63.6%

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

      1. *-commutative 63.6%

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

   10. Simplified 63.6%

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

   if -2.25000000000000014e-5 < x < 2.0000000000000001e-4

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

   if 2.0000000000000001e-4 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 60.5%

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

   4. Taylor expanded in x around inf 60.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0002:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)\right)}{\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \left(3 + \left(\sqrt{5} \cdot 1.5 + -1.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sqrt{2} \cdot \left(\sin y - \sin x \cdot 0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \cos x \cdot \left(t\_0 + -0.5\right)\\ t_2 := \cos x - \cos y\\ t_3 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2} + y \cdot \left(\sin x \cdot 1.00390625\right)\right)\right)}{3 + 3 \cdot \left(t\_1 + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t\_2 \cdot \left(-0.0625 \cdot t\_3\right)\right)}{3 + 3 \cdot \left(t\_1 + \cos y \cdot \left(1.5 - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (* (cos x) (+ t_0 -0.5)))
        (t_2 (- (cos x) (cos y)))
        (t_3 (pow (sin y) 2.0)))
   (if (<= y -0.039)
     (/
      (+ 2.0 (* t_2 (* t_3 (* (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)))))
     (if (<= y 0.0135)
       (/
        (+
         2.0
         (*
          (+ (cos x) -1.0)
          (*
           (sqrt 2.0)
           (+ (* -0.0625 (pow (sin x) 2.0)) (* y (* (sin x) 1.00390625))))))
        (+ 3.0 (* 3.0 (+ t_1 (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5))))))))
       (/
        (+ 2.0 (* (sqrt 2.0) (* t_2 (* -0.0625 t_3))))
        (+ 3.0 (* 3.0 (+ t_1 (* (cos y) (- 1.5 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = cos(x) * (t_0 + -0.5);
	double t_2 = cos(x) - cos(y);
	double t_3 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * -0.0625)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (y <= 0.0135) {
		tmp = (2.0 + ((cos(x) + -1.0) * (sqrt(2.0) * ((-0.0625 * pow(sin(x), 2.0)) + (y * (sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (t_1 + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * (t_1 + (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 = cos(x) * (t_0 + (-0.5d0))
    t_2 = cos(x) - cos(y)
    t_3 = sin(y) ** 2.0d0
    if (y <= (-0.039d0)) then
        tmp = (2.0d0 + (t_2 * (t_3 * (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))))
    else if (y <= 0.0135d0) then
        tmp = (2.0d0 + ((cos(x) + (-1.0d0)) * (sqrt(2.0d0) * (((-0.0625d0) * (sin(x) ** 2.0d0)) + (y * (sin(x) * 1.00390625d0)))))) / (3.0d0 + (3.0d0 * (t_1 + (cos(y) * (1.0d0 / (1.5d0 + (sqrt(5.0d0) * 0.5d0)))))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_2 * ((-0.0625d0) * t_3)))) / (3.0d0 + (3.0d0 * (t_1 + (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 = Math.cos(x) * (t_0 + -0.5);
	double t_2 = Math.cos(x) - Math.cos(y);
	double t_3 = Math.pow(Math.sin(y), 2.0);
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * (t_3 * (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))));
	} else if (y <= 0.0135) {
		tmp = (2.0 + ((Math.cos(x) + -1.0) * (Math.sqrt(2.0) * ((-0.0625 * Math.pow(Math.sin(x), 2.0)) + (y * (Math.sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (t_1 + (Math.cos(y) * (1.0 / (1.5 + (Math.sqrt(5.0) * 0.5)))))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * (t_1 + (Math.cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.cos(x) * (t_0 + -0.5)
	t_2 = math.cos(x) - math.cos(y)
	t_3 = math.pow(math.sin(y), 2.0)
	tmp = 0
	if y <= -0.039:
		tmp = (2.0 + (t_2 * (t_3 * (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))))
	elif y <= 0.0135:
		tmp = (2.0 + ((math.cos(x) + -1.0) * (math.sqrt(2.0) * ((-0.0625 * math.pow(math.sin(x), 2.0)) + (y * (math.sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (t_1 + (math.cos(y) * (1.0 / (1.5 + (math.sqrt(5.0) * 0.5)))))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * (t_1 + (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(cos(x) * Float64(t_0 + -0.5))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.039)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * 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)))));
	elseif (y <= 0.0135)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) + Float64(y * Float64(sin(x) * 1.00390625)))))) / Float64(3.0 + Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_2 * Float64(-0.0625 * t_3)))) / Float64(3.0 + Float64(3.0 * Float64(t_1 + 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 = cos(x) * (t_0 + -0.5);
	t_2 = cos(x) - cos(y);
	t_3 = sin(y) ^ 2.0;
	tmp = 0.0;
	if (y <= -0.039)
		tmp = (2.0 + (t_2 * (t_3 * (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))));
	elseif (y <= 0.0135)
		tmp = (2.0 + ((cos(x) + -1.0) * (sqrt(2.0) * ((-0.0625 * (sin(x) ^ 2.0)) + (y * (sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (t_1 + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	else
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * (t_1 + (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[(N[Cos[x], $MachinePrecision] * N[(t$95$0 + -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.039], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * 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], If[LessEqual[y, 0.0135], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[Sin[x], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(-0.0625 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0389999999999999999

   1. Initial program 99.2%

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

      1. flip-- 26.6%

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

      2. metadata-eval 26.6%

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

      3. pow1/2 26.6%

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

      4. pow1/2 26.6%

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

      5. pow-prod-up 26.6%

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

      6. metadata-eval 26.6%

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

      7. metadata-eval 26.6%

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

      8. metadata-eval 26.6%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

   7. Simplified 59.4%

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

   if -0.0389999999999999999 < y < 0.0134999999999999998

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow2 99.5%

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

      4. div-inv 99.5%

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

      5. metadata-eval 99.5%

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

      6. div-inv 99.5%

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

      7. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. neg-sub0 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{\left(0 - {\left(\sqrt{5} \cdot 0.5\right)}^{2}\right)} + 2.25}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      4. associate-+l- 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{0 - \left({\left(\sqrt{5} \cdot 0.5\right)}^{2} - 2.25\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      5. unpow2 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      6. swap-sqr 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      7. rem-square-sqrt 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right) - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(5 \cdot \color{blue}{0.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      9. metadata-eval 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{1.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      10. metadata-eval 99.6%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \color{blue}{-1}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 98.4%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + 0.5 \cdot \sqrt{5}}\right)} \]
   10. Step-by-step derivation

       1. sub-neg 97.7%

          \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right) + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       2. metadata-eval 97.7%

          \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right) + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       3. associate-*r* 97.7%

          \[\leadsto \frac{2 + \left(-0.0625 \cdot \color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x + -1\right)\right)} + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       4. associate-*r* 97.7%

          \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right)} + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       5. associate-*r* 97.7%

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \color{blue}{\left(y \cdot \sqrt{2}\right) \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)}\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       6. +-commutative 97.7%

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \left(\cos x - 1\right) + 0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)\right)}\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       7. distribute-rgt1-in 97.7%

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)\right)}\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       8. sub-neg 97.7%

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \left(\left(0.00390625 + 1\right) \cdot \left(\sin x \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

       9. metadata-eval 97.7%

          \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \left(\left(0.00390625 + 1\right) \cdot \left(\sin x \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

   11. Simplified 98.4%

       \[\leadsto \frac{2 + \color{blue}{\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2} + \left(\sin x \cdot 1.00390625\right) \cdot y\right)\right)}}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + 0.5 \cdot \sqrt{5}}\right)} \]

   if 0.0134999999999999998 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 62.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2} + y \cdot \left(\sin x \cdot 1.00390625\right)\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ t_1 := \cos x - \cos y\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t\_0}\\ \mathbf{elif}\;y \leq 4600000:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t\_1 \cdot \left(-0.0625 \cdot t\_3\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(t\_2 + -0.5\right) + \cos y \cdot \left(1.5 - t\_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
        (t_1 (- (cos x) (cos y)))
        (t_2 (/ (sqrt 5.0) 2.0))
        (t_3 (pow (sin y) 2.0)))
   (if (<= y -0.039)
     (/ (+ 2.0 (* t_1 (* t_3 (* (sqrt 2.0) -0.0625)))) t_0)
     (if (<= y 4600000.0)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
          (+ (cos x) -1.0)))
        t_0)
       (/
        (+ 2.0 (* (sqrt 2.0) (* t_1 (* -0.0625 t_3))))
        (+
         3.0
         (* 3.0 (+ (* (cos x) (+ t_2 -0.5)) (* (cos y) (- 1.5 t_2))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	double t_1 = cos(x) - cos(y);
	double t_2 = sqrt(5.0) / 2.0;
	double t_3 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_1 * (t_3 * (sqrt(2.0) * -0.0625)))) / t_0;
	} else if (y <= 4600000.0) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / t_0;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_1 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((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) :: t_3
    real(8) :: tmp
    t_0 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0)))
    t_1 = cos(x) - cos(y)
    t_2 = sqrt(5.0d0) / 2.0d0
    t_3 = sin(y) ** 2.0d0
    if (y <= (-0.039d0)) then
        tmp = (2.0d0 + (t_1 * (t_3 * (sqrt(2.0d0) * (-0.0625d0))))) / t_0
    else if (y <= 4600000.0d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) + (-1.0d0)))) / t_0
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_1 * ((-0.0625d0) * t_3)))) / (3.0d0 + (3.0d0 * ((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 = 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 t_1 = Math.cos(x) - Math.cos(y);
	double t_2 = Math.sqrt(5.0) / 2.0;
	double t_3 = Math.pow(Math.sin(y), 2.0);
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_1 * (t_3 * (Math.sqrt(2.0) * -0.0625)))) / t_0;
	} else if (y <= 4600000.0) {
		tmp = (2.0 + (((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) + -1.0))) / t_0;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_1 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((Math.cos(x) * (t_2 + -0.5)) + (Math.cos(y) * (1.5 - t_2)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0)))
	t_1 = math.cos(x) - math.cos(y)
	t_2 = math.sqrt(5.0) / 2.0
	t_3 = math.pow(math.sin(y), 2.0)
	tmp = 0
	if y <= -0.039:
		tmp = (2.0 + (t_1 * (t_3 * (math.sqrt(2.0) * -0.0625)))) / t_0
	elif y <= 4600000.0:
		tmp = (2.0 + (((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) + -1.0))) / t_0
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_1 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((math.cos(x) * (t_2 + -0.5)) + (math.cos(y) * (1.5 - t_2)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(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))))
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(sqrt(5.0) / 2.0)
	t_3 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.039)
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(t_3 * Float64(sqrt(2.0) * -0.0625)))) / t_0);
	elseif (y <= 4600000.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) + -1.0))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_1 * Float64(-0.0625 * t_3)))) / Float64(3.0 + Float64(3.0 * Float64(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 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	t_1 = cos(x) - cos(y);
	t_2 = sqrt(5.0) / 2.0;
	t_3 = sin(y) ^ 2.0;
	tmp = 0.0;
	if (y <= -0.039)
		tmp = (2.0 + (t_1 * (t_3 * (sqrt(2.0) * -0.0625)))) / t_0;
	elseif (y <= 4600000.0)
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / t_0;
	else
		tmp = (2.0 + (sqrt(2.0) * (t_1 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((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[(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]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.039], N[(N[(2.0 + N[(t$95$1 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 4600000.0], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$1 * N[(-0.0625 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * 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]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0389999999999999999

   1. Initial program 99.2%

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

      1. flip-- 26.6%

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

      2. metadata-eval 26.6%

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

      3. pow1/2 26.6%

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

      4. pow1/2 26.6%

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

      5. pow-prod-up 26.6%

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

      6. metadata-eval 26.6%

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

      7. metadata-eval 26.6%

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

      8. metadata-eval 26.6%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

   7. Simplified 59.4%

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

   if -0.0389999999999999999 < y < 4.6e6

   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-- 97.2%

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

      2. metadata-eval 97.2%

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

      3. pow1/2 97.2%

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

      4. pow1/2 97.2%

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

      5. pow-prod-up 97.4%

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

      6. metadata-eval 97.4%

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

      7. metadata-eval 97.4%

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

      8. metadata-eval 97.4%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0 97.7%

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

   6. Taylor expanded in y around 0 97.7%

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

   if 4.6e6 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 62.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 4600000:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := \cos x - \cos y\\ t_3 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(t\_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 4600000:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(t\_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t\_2 \cdot \left(-0.0625 \cdot t\_3\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(t\_0 + -0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (- (cos x) (cos y)))
        (t_3 (pow (sin y) 2.0)))
   (if (<= y -0.039)
     (/
      (+ 2.0 (* t_2 (* t_3 (* (sqrt 2.0) -0.0625))))
      (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= y 4600000.0)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
          (+ (cos x) -1.0)))
        (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
       (/
        (+ 2.0 (* (sqrt 2.0) (* t_2 (* -0.0625 t_3))))
        (+
         3.0
         (* 3.0 (+ (* (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 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = cos(x) - cos(y);
	double t_3 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * -0.0625)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (y <= 4600000.0) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((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 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_2 = cos(x) - cos(y)
    t_3 = sin(y) ** 2.0d0
    if (y <= (-0.039d0)) then
        tmp = (2.0d0 + (t_2 * (t_3 * (sqrt(2.0d0) * (-0.0625d0))))) / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else if (y <= 4600000.0d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) + (-1.0d0)))) / (3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_2 * ((-0.0625d0) * t_3)))) / (3.0d0 + (3.0d0 * ((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 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_2 = Math.cos(x) - Math.cos(y);
	double t_3 = Math.pow(Math.sin(y), 2.0);
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * (t_3 * (Math.sqrt(2.0) * -0.0625)))) / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else if (y <= 4600000.0) {
		tmp = (2.0 + (((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) + -1.0))) / (3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((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 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_2 = math.cos(x) - math.cos(y)
	t_3 = math.pow(math.sin(y), 2.0)
	tmp = 0
	if y <= -0.039:
		tmp = (2.0 + (t_2 * (t_3 * (math.sqrt(2.0) * -0.0625)))) / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	elif y <= 4600000.0:
		tmp = (2.0 + (((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) + -1.0))) / (3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((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(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(cos(x) - cos(y))
	t_3 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -0.039)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sqrt(2.0) * -0.0625)))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (y <= 4600000.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_2 * Float64(-0.0625 * t_3)))) / Float64(3.0 + Float64(3.0 * Float64(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 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_2 = cos(x) - cos(y);
	t_3 = sin(y) ^ 2.0;
	tmp = 0.0;
	if (y <= -0.039)
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * -0.0625)))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	elseif (y <= 4600000.0)
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	else
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (-0.0625 * t_3)))) / (3.0 + (3.0 * ((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[(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]}, Block[{t$95$3 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -0.039], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4600000.0], N[(N[(2.0 + N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $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], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(-0.0625 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * 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. Split input into 3 regimes

2. if y < -0.0389999999999999999

   1. Initial program 99.2%

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

      1. flip-- 26.6%

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

      2. metadata-eval 26.6%

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

      3. pow1/2 26.6%

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

      4. pow1/2 26.6%

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

      5. pow-prod-up 26.6%

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

      6. metadata-eval 26.6%

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

      7. metadata-eval 26.6%

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

      8. metadata-eval 26.6%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

   7. Simplified 59.4%

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

   if -0.0389999999999999999 < y < 4.6e6

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0 97.7%

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

   4. Taylor expanded in y around 0 97.7%

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

   if 4.6e6 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 62.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 4600000:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{3 + \sqrt{5}}\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := {\sin y}^{2}\\ t_3 := t\_1 + -0.5\\ \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot t\_3\right)\right) + 1.5 \cdot \left(\cos y \cdot t\_0\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t\_2\right)\right)}{3 + 3 \cdot \left(\cos x \cdot t\_3 + \cos y \cdot \left(1.5 - t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 4.0 (+ 3.0 (sqrt 5.0))))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2 (pow (sin y) 2.0))
        (t_3 (+ t_1 -0.5)))
   (if (<= y -0.039)
     (/
      (+ 2.0 (* t_2 (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625))))
      (+ (+ 3.0 (* (cos x) (* 3.0 t_3))) (* 1.5 (* (cos y) t_0))))
     (if (<= y 0.0135)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (* (+ (cos x) -1.0) (* (sin x) (+ y (* (sin x) -0.0625))))))
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
          (* (cos y) (/ t_0 2.0)))))
       (/
        (+ 2.0 (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 t_2))))
        (+ 3.0 (* 3.0 (+ (* (cos x) t_3) (* (cos y) (- 1.5 t_1))))))))))

C

double code(double x, double y) {
	double t_0 = 4.0 / (3.0 + sqrt(5.0));
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = pow(sin(y), 2.0);
	double t_3 = t_1 + -0.5;
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * t_3))) + (1.5 * (cos(y) * t_0)));
	} else if (y <= 0.0135) {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) * (y + (sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_2)))) / (3.0 + (3.0 * ((cos(x) * t_3) + (cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 4.0d0 / (3.0d0 + sqrt(5.0d0))
    t_1 = sqrt(5.0d0) / 2.0d0
    t_2 = sin(y) ** 2.0d0
    t_3 = t_1 + (-0.5d0)
    if (y <= (-0.039d0)) then
        tmp = (2.0d0 + (t_2 * ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (-0.0625d0))))) / ((3.0d0 + (cos(x) * (3.0d0 * t_3))) + (1.5d0 * (cos(y) * t_0)))
    else if (y <= 0.0135d0) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) * (y + (sin(x) * (-0.0625d0))))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * (t_0 / 2.0d0))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((-0.0625d0) * t_2)))) / (3.0d0 + (3.0d0 * ((cos(x) * t_3) + (cos(y) * (1.5d0 - t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 4.0 / (3.0 + Math.sqrt(5.0));
	double t_1 = Math.sqrt(5.0) / 2.0;
	double t_2 = Math.pow(Math.sin(y), 2.0);
	double t_3 = t_1 + -0.5;
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * -0.0625)))) / ((3.0 + (Math.cos(x) * (3.0 * t_3))) + (1.5 * (Math.cos(y) * t_0)));
	} else if (y <= 0.0135) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * (Math.sin(x) * (y + (Math.sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * (t_0 / 2.0))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (-0.0625 * t_2)))) / (3.0 + (3.0 * ((Math.cos(x) * t_3) + (Math.cos(y) * (1.5 - t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 4.0 / (3.0 + math.sqrt(5.0))
	t_1 = math.sqrt(5.0) / 2.0
	t_2 = math.pow(math.sin(y), 2.0)
	t_3 = t_1 + -0.5
	tmp = 0
	if y <= -0.039:
		tmp = (2.0 + (t_2 * ((1.0 - math.cos(y)) * (math.sqrt(2.0) * -0.0625)))) / ((3.0 + (math.cos(x) * (3.0 * t_3))) + (1.5 * (math.cos(y) * t_0)))
	elif y <= 0.0135:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) + -1.0) * (math.sin(x) * (y + (math.sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * (t_0 / 2.0))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (-0.0625 * t_2)))) / (3.0 + (3.0 * ((math.cos(x) * t_3) + (math.cos(y) * (1.5 - t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(4.0 / Float64(3.0 + sqrt(5.0)))
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = sin(y) ^ 2.0
	t_3 = Float64(t_1 + -0.5)
	tmp = 0.0
	if (y <= -0.039)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)))) / Float64(Float64(3.0 + Float64(cos(x) * Float64(3.0 * t_3))) + Float64(1.5 * Float64(cos(y) * t_0))));
	elseif (y <= 0.0135)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * Float64(sin(x) * Float64(y + Float64(sin(x) * -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(t_0 / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * t_2)))) / Float64(3.0 + Float64(3.0 * Float64(Float64(cos(x) * t_3) + Float64(cos(y) * Float64(1.5 - t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 4.0 / (3.0 + sqrt(5.0));
	t_1 = sqrt(5.0) / 2.0;
	t_2 = sin(y) ^ 2.0;
	t_3 = t_1 + -0.5;
	tmp = 0.0;
	if (y <= -0.039)
		tmp = (2.0 + (t_2 * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * t_3))) + (1.5 * (cos(y) * t_0)));
	elseif (y <= 0.0135)
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) * (y + (sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_2)))) / (3.0 + (3.0 * ((cos(x) * t_3) + (cos(y) * (1.5 - t_1)))));
	end
	tmp_2 = 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[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + -0.5), $MachinePrecision]}, If[LessEqual[y, -0.039], N[(N[(2.0 + N[(t$95$2 * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0135], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(y + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $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[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0389999999999999999

   1. Initial program 99.2%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 59.3%

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

      1. *-commutative 59.3%

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

      2. associate-*l* 59.3%

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

      3. associate-*r* 59.3%

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

      4. *-commutative 59.3%

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

      5. *-commutative 59.3%

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

      6. *-commutative 59.3%

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

      7. associate-*l* 59.3%

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

   7. Simplified 59.3%

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

      1. flip-- 26.6%

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

      2. metadata-eval 26.6%

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

      3. pow1/2 26.6%

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

      4. pow1/2 26.6%

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

      5. pow-prod-up 26.6%

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

      6. metadata-eval 26.6%

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

      7. metadata-eval 26.6%

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

      8. metadata-eval 26.6%

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

   9. Applied egg-rr 59.3%

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

   if -0.0389999999999999999 < y < 0.0134999999999999998

   1. Initial program 99.5%

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

      1. flip-- 97.8%

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

      2. metadata-eval 97.8%

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

      3. pow1/2 97.8%

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

      4. pow1/2 97.8%

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

      5. pow-prod-up 98.0%

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

      6. metadata-eval 98.0%

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

      7. metadata-eval 98.0%

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

      8. metadata-eval 98.0%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0 98.2%

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

   6. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-*r* 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*r* 98.2%

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

      5. associate-*r* 98.2%

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

      6. associate-*r* 98.2%

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

      7. *-commutative 98.2%

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

      8. associate-*r* 98.2%

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

      9. associate-*r* 98.2%

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

      10. distribute-rgt-out 98.2%

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

   8. Simplified 98.2%

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

   if 0.0134999999999999998 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 62.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ t_1 := {\sin y}^{2}\\ t_2 := \cos x - \cos y\\ t_3 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t\_0}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t\_2 \cdot \left(-0.0625 \cdot t\_1\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(t\_3 + -0.5\right) + \cos y \cdot \left(1.5 - t\_3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
        (t_1 (pow (sin y) 2.0))
        (t_2 (- (cos x) (cos y)))
        (t_3 (/ (sqrt 5.0) 2.0)))
   (if (<= y -0.039)
     (/ (+ 2.0 (* t_2 (* t_1 (* (sqrt 2.0) -0.0625)))) t_0)
     (if (<= y 0.0135)
       (/
        (+
         2.0
         (*
          (sqrt 2.0)
          (* (+ (cos x) -1.0) (* (sin x) (+ y (* (sin x) -0.0625))))))
        t_0)
       (/
        (+ 2.0 (* (sqrt 2.0) (* t_2 (* -0.0625 t_1))))
        (+
         3.0
         (* 3.0 (+ (* (cos x) (+ t_3 -0.5)) (* (cos y) (- 1.5 t_3))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	double t_1 = pow(sin(y), 2.0);
	double t_2 = cos(x) - cos(y);
	double t_3 = sqrt(5.0) / 2.0;
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * (t_1 * (sqrt(2.0) * -0.0625)))) / t_0;
	} else if (y <= 0.0135) {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) * (y + (sin(x) * -0.0625)))))) / t_0;
	} else {
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (-0.0625 * t_1)))) / (3.0 + (3.0 * ((cos(x) * (t_3 + -0.5)) + (cos(y) * (1.5 - 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 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0)))
    t_1 = sin(y) ** 2.0d0
    t_2 = cos(x) - cos(y)
    t_3 = sqrt(5.0d0) / 2.0d0
    if (y <= (-0.039d0)) then
        tmp = (2.0d0 + (t_2 * (t_1 * (sqrt(2.0d0) * (-0.0625d0))))) / t_0
    else if (y <= 0.0135d0) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) * (y + (sin(x) * (-0.0625d0))))))) / t_0
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_2 * ((-0.0625d0) * t_1)))) / (3.0d0 + (3.0d0 * ((cos(x) * (t_3 + (-0.5d0))) + (cos(y) * (1.5d0 - t_3)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0)));
	double t_1 = Math.pow(Math.sin(y), 2.0);
	double t_2 = Math.cos(x) - Math.cos(y);
	double t_3 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if (y <= -0.039) {
		tmp = (2.0 + (t_2 * (t_1 * (Math.sqrt(2.0) * -0.0625)))) / t_0;
	} else if (y <= 0.0135) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * (Math.sin(x) * (y + (Math.sin(x) * -0.0625)))))) / t_0;
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_2 * (-0.0625 * t_1)))) / (3.0 + (3.0 * ((Math.cos(x) * (t_3 + -0.5)) + (Math.cos(y) * (1.5 - t_3)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0)))
	t_1 = math.pow(math.sin(y), 2.0)
	t_2 = math.cos(x) - math.cos(y)
	t_3 = math.sqrt(5.0) / 2.0
	tmp = 0
	if y <= -0.039:
		tmp = (2.0 + (t_2 * (t_1 * (math.sqrt(2.0) * -0.0625)))) / t_0
	elif y <= 0.0135:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) + -1.0) * (math.sin(x) * (y + (math.sin(x) * -0.0625)))))) / t_0
	else:
		tmp = (2.0 + (math.sqrt(2.0) * (t_2 * (-0.0625 * t_1)))) / (3.0 + (3.0 * ((math.cos(x) * (t_3 + -0.5)) + (math.cos(y) * (1.5 - t_3)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0))))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(cos(x) - cos(y))
	t_3 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (y <= -0.039)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_1 * Float64(sqrt(2.0) * -0.0625)))) / t_0);
	elseif (y <= 0.0135)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * Float64(sin(x) * Float64(y + Float64(sin(x) * -0.0625)))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_2 * Float64(-0.0625 * t_1)))) / Float64(3.0 + Float64(3.0 * Float64(Float64(cos(x) * Float64(t_3 + -0.5)) + Float64(cos(y) * Float64(1.5 - t_3))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	t_1 = sin(y) ^ 2.0;
	t_2 = cos(x) - cos(y);
	t_3 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if (y <= -0.039)
		tmp = (2.0 + (t_2 * (t_1 * (sqrt(2.0) * -0.0625)))) / t_0;
	elseif (y <= 0.0135)
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) * (y + (sin(x) * -0.0625)))))) / t_0;
	else
		tmp = (2.0 + (sqrt(2.0) * (t_2 * (-0.0625 * t_1)))) / (3.0 + (3.0 * ((cos(x) * (t_3 + -0.5)) + (cos(y) * (1.5 - t_3)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[y, -0.039], N[(N[(2.0 + N[(t$95$2 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 0.0135], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(y + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(-0.0625 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$3 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0389999999999999999

   1. Initial program 99.2%

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

      1. flip-- 26.6%

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

      2. metadata-eval 26.6%

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

      3. pow1/2 26.6%

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

      4. pow1/2 26.6%

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

      5. pow-prod-up 26.6%

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

      6. metadata-eval 26.6%

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

      7. metadata-eval 26.6%

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

      8. metadata-eval 26.6%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

   7. Simplified 59.4%

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

   if -0.0389999999999999999 < y < 0.0134999999999999998

   1. Initial program 99.5%

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

      1. flip-- 97.8%

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

      2. metadata-eval 97.8%

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

      3. pow1/2 97.8%

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

      4. pow1/2 97.8%

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

      5. pow-prod-up 98.0%

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

      6. metadata-eval 98.0%

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

      7. metadata-eval 98.0%

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

      8. metadata-eval 98.0%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0 98.2%

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

   6. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-*r* 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*r* 98.2%

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

      5. associate-*r* 98.2%

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

      6. associate-*r* 98.2%

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

      7. *-commutative 98.2%

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

      8. associate-*r* 98.2%

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

      9. associate-*r* 98.2%

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

      10. distribute-rgt-out 98.2%

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

   8. Simplified 98.2%

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

   if 0.0134999999999999998 < y

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around 0 62.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{3 + \sqrt{5}}\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 0.0155\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{t\_0}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 4.0 (+ 3.0 (sqrt 5.0)))))
   (if (or (<= y -0.039) (not (<= y 0.0155)))
     (/
      (+ 2.0 (* (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625))))
      (+
       (+ 3.0 (* (cos x) (* 3.0 (+ (/ (sqrt 5.0) 2.0) -0.5))))
       (* 1.5 (* (cos y) t_0))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (* (+ (cos x) -1.0) (* (sin x) (+ y (* (sin x) -0.0625))))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ t_0 2.0))))))))

C

double code(double x, double y) {
	double t_0 = 4.0 / (3.0 + sqrt(5.0));
	double tmp;
	if ((y <= -0.039) || !(y <= 0.0155)) {
		tmp = (2.0 + (pow(sin(y), 2.0) * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (cos(y) * t_0)));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) * (y + (sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_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) :: tmp
    t_0 = 4.0d0 / (3.0d0 + sqrt(5.0d0))
    if ((y <= (-0.039d0)) .or. (.not. (y <= 0.0155d0))) then
        tmp = (2.0d0 + ((sin(y) ** 2.0d0) * ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (-0.0625d0))))) / ((3.0d0 + (cos(x) * (3.0d0 * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0))))) + (1.5d0 * (cos(y) * t_0)))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) * (y + (sin(x) * (-0.0625d0))))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * (t_0 / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 4.0 / (3.0 + Math.sqrt(5.0));
	double tmp;
	if ((y <= -0.039) || !(y <= 0.0155)) {
		tmp = (2.0 + (Math.pow(Math.sin(y), 2.0) * ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * -0.0625)))) / ((3.0 + (Math.cos(x) * (3.0 * ((Math.sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (Math.cos(y) * t_0)));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * (Math.sin(x) * (y + (Math.sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * (t_0 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 4.0 / (3.0 + math.sqrt(5.0))
	tmp = 0
	if (y <= -0.039) or not (y <= 0.0155):
		tmp = (2.0 + (math.pow(math.sin(y), 2.0) * ((1.0 - math.cos(y)) * (math.sqrt(2.0) * -0.0625)))) / ((3.0 + (math.cos(x) * (3.0 * ((math.sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (math.cos(y) * t_0)))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) + -1.0) * (math.sin(x) * (y + (math.sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * (t_0 / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(4.0 / Float64(3.0 + sqrt(5.0)))
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 0.0155))
		tmp = Float64(Float64(2.0 + Float64((sin(y) ^ 2.0) * Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)))) / Float64(Float64(3.0 + Float64(cos(x) * Float64(3.0 * Float64(Float64(sqrt(5.0) / 2.0) + -0.5)))) + Float64(1.5 * Float64(cos(y) * t_0))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * Float64(sin(x) * Float64(y + Float64(sin(x) * -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(t_0 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 4.0 / (3.0 + sqrt(5.0));
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 0.0155)))
		tmp = (2.0 + ((sin(y) ^ 2.0) * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (cos(y) * t_0)));
	else
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) * (y + (sin(x) * -0.0625)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * (t_0 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 0.0155]], $MachinePrecision]], N[(N[(2.0 + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(y + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $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[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 0.0155 < y

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. associate-*l* 60.5%

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

      3. associate-*r* 60.5%

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

      4. *-commutative 60.5%

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

      5. *-commutative 60.5%

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

      6. *-commutative 60.5%

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

      7. associate-*l* 60.5%

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

   7. Simplified 60.5%

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

      1. flip-- 26.8%

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

      2. metadata-eval 26.8%

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

      3. pow1/2 26.8%

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

      4. pow1/2 26.8%

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

      5. pow-prod-up 26.8%

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

      6. metadata-eval 26.8%

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

      7. metadata-eval 26.8%

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

      8. metadata-eval 26.8%

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

   9. Applied egg-rr 60.6%

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

   if -0.0389999999999999999 < y < 0.0155

   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-- 97.8%

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

      2. metadata-eval 97.8%

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

      3. pow1/2 97.8%

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

      4. pow1/2 97.8%

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

      5. pow-prod-up 98.0%

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

      6. metadata-eval 98.0%

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

      7. metadata-eval 98.0%

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

      8. metadata-eval 98.0%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0 98.2%

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

   6. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-*r* 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*r* 98.2%

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

      5. associate-*r* 98.2%

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

      6. associate-*r* 98.2%

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

      7. *-commutative 98.2%

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

      8. associate-*r* 98.2%

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

      9. associate-*r* 98.2%

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

      10. distribute-rgt-out 98.2%

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

   8. Simplified 98.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 0.0155\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x \cdot \left(y + \sin x \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2} + y \cdot \left(\sin x \cdot 1.00390625\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + t\_0\right) - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+ 2.0 (* (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625))))
      (+
       (+ 3.0 (* (cos x) (* 3.0 (+ (/ (sqrt 5.0) 2.0) -0.5))))
       (* 1.5 (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))))))
     (/
      (+
       2.0
       (*
        (+ (cos x) -1.0)
        (*
         (sqrt 2.0)
         (+ (* -0.0625 (pow (sin x) 2.0)) (* y (* (sin x) 1.00390625))))))
      (+ 3.0 (* 3.0 (+ 1.5 (- (* (cos x) (+ -0.5 t_0)) t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (pow(sin(y), 2.0) * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (cos(y) * (4.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = (2.0 + ((cos(x) + -1.0) * (sqrt(2.0) * ((-0.0625 * pow(sin(x), 2.0)) + (y * (sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_0)) - t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + ((sin(y) ** 2.0d0) * ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (-0.0625d0))))) / ((3.0d0 + (cos(x) * (3.0d0 * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0))))) + (1.5d0 * (cos(y) * (4.0d0 / (3.0d0 + sqrt(5.0d0))))))
    else
        tmp = (2.0d0 + ((cos(x) + (-1.0d0)) * (sqrt(2.0d0) * (((-0.0625d0) * (sin(x) ** 2.0d0)) + (y * (sin(x) * 1.00390625d0)))))) / (3.0d0 + (3.0d0 * (1.5d0 + ((cos(x) * ((-0.5d0) + t_0)) - t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (Math.pow(Math.sin(y), 2.0) * ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * -0.0625)))) / ((3.0 + (Math.cos(x) * (3.0 * ((Math.sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (Math.cos(y) * (4.0 / (3.0 + Math.sqrt(5.0))))));
	} else {
		tmp = (2.0 + ((Math.cos(x) + -1.0) * (Math.sqrt(2.0) * ((-0.0625 * Math.pow(Math.sin(x), 2.0)) + (y * (Math.sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((Math.cos(x) * (-0.5 + t_0)) - t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (math.pow(math.sin(y), 2.0) * ((1.0 - math.cos(y)) * (math.sqrt(2.0) * -0.0625)))) / ((3.0 + (math.cos(x) * (3.0 * ((math.sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (math.cos(y) * (4.0 / (3.0 + math.sqrt(5.0))))))
	else:
		tmp = (2.0 + ((math.cos(x) + -1.0) * (math.sqrt(2.0) * ((-0.0625 * math.pow(math.sin(x), 2.0)) + (y * (math.sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((math.cos(x) * (-0.5 + t_0)) - t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64((sin(y) ^ 2.0) * Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)))) / Float64(Float64(3.0 + Float64(cos(x) * Float64(3.0 * Float64(Float64(sqrt(5.0) / 2.0) + -0.5)))) + Float64(1.5 * Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) + Float64(y * Float64(sin(x) * 1.00390625)))))) / Float64(3.0 + Float64(3.0 * Float64(1.5 + Float64(Float64(cos(x) * Float64(-0.5 + t_0)) - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + ((sin(y) ^ 2.0) * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (cos(y) * (4.0 / (3.0 + sqrt(5.0))))));
	else
		tmp = (2.0 + ((cos(x) + -1.0) * (sqrt(2.0) * ((-0.0625 * (sin(x) ^ 2.0)) + (y * (sin(x) * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_0)) - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[Sin[x], $MachinePrecision] * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(1.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.1%

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

   5. Taylor expanded in x around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. *-commutative 61.8%

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

      6. *-commutative 61.8%

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

      7. associate-*l* 61.8%

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

   7. Simplified 61.8%

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

      1. flip-- 29.5%

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

      2. metadata-eval 29.5%

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

      3. pow1/2 29.5%

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

      4. pow1/2 29.5%

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

      5. pow-prod-up 29.5%

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

      6. metadata-eval 29.5%

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

      7. metadata-eval 29.5%

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

      8. metadata-eval 29.5%

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

   9. Applied egg-rr 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. associate--l+ 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around 0 98.6%

      \[\leadsto \frac{2 + \color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x - 1\right)\right)\right) + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. sub-neg 98.6%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right) + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      2. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right) + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      3. associate-*r* 98.6%

         \[\leadsto \frac{2 + \left(-0.0625 \cdot \color{blue}{\left(\left({\sin x}^{2} \cdot \sqrt{2}\right) \cdot \left(\cos x + -1\right)\right)} + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      4. associate-*r* 98.6%

         \[\leadsto \frac{2 + \left(\color{blue}{\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right)} + y \cdot \left(\sqrt{2} \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      5. associate-*r* 98.6%

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \color{blue}{\left(y \cdot \sqrt{2}\right) \cdot \left(0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right) + \sin x \cdot \left(\cos x - 1\right)\right)}\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      6. +-commutative 98.6%

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\sin x \cdot \left(\cos x - 1\right) + 0.00390625 \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)\right)}\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      7. distribute-rgt1-in 98.6%

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \left(\sin x \cdot \left(\cos x - 1\right)\right)\right)}\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      8. sub-neg 98.6%

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \left(\left(0.00390625 + 1\right) \cdot \left(\sin x \cdot \color{blue}{\left(\cos x + \left(-1\right)\right)}\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      9. metadata-eval 98.6%

         \[\leadsto \frac{2 + \left(\left(-0.0625 \cdot \left({\sin x}^{2} \cdot \sqrt{2}\right)\right) \cdot \left(\cos x + -1\right) + \left(y \cdot \sqrt{2}\right) \cdot \left(\left(0.00390625 + 1\right) \cdot \left(\sin x \cdot \left(\cos x + \color{blue}{-1}\right)\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

   10. Simplified 98.6%

       \[\leadsto \frac{2 + \color{blue}{\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2} + \left(\sin x \cdot 1.00390625\right) \cdot y\right)\right)}}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin x}^{2} + y \cdot \left(\sin x \cdot 1.00390625\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin x \cdot -0.0625 + y \cdot 1.00390625\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + t\_0\right) - t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+ 2.0 (* (pow (sin y) 2.0) (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625))))
      (+
       (+ 3.0 (* (cos x) (* 3.0 (+ (/ (sqrt 5.0) 2.0) -0.5))))
       (* 1.5 (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))))))
     (/
      (+
       2.0
       (*
        (sqrt 2.0)
        (*
         (- (cos x) (cos y))
         (* (sin x) (+ (* (sin x) -0.0625) (* y 1.00390625))))))
      (+ 3.0 (* 3.0 (+ 1.5 (- (* (cos x) (+ -0.5 t_0)) t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (pow(sin(y), 2.0) * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (cos(y) * (4.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (sin(x) * ((sin(x) * -0.0625) + (y * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_0)) - t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + ((sin(y) ** 2.0d0) * ((1.0d0 - cos(y)) * (sqrt(2.0d0) * (-0.0625d0))))) / ((3.0d0 + (cos(x) * (3.0d0 * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0))))) + (1.5d0 * (cos(y) * (4.0d0 / (3.0d0 + sqrt(5.0d0))))))
    else
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * (sin(x) * ((sin(x) * (-0.0625d0)) + (y * 1.00390625d0)))))) / (3.0d0 + (3.0d0 * (1.5d0 + ((cos(x) * ((-0.5d0) + t_0)) - t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (Math.pow(Math.sin(y), 2.0) * ((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * -0.0625)))) / ((3.0 + (Math.cos(x) * (3.0 * ((Math.sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (Math.cos(y) * (4.0 / (3.0 + Math.sqrt(5.0))))));
	} else {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (Math.sin(x) * ((Math.sin(x) * -0.0625) + (y * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((Math.cos(x) * (-0.5 + t_0)) - t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (math.pow(math.sin(y), 2.0) * ((1.0 - math.cos(y)) * (math.sqrt(2.0) * -0.0625)))) / ((3.0 + (math.cos(x) * (3.0 * ((math.sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (math.cos(y) * (4.0 / (3.0 + math.sqrt(5.0))))))
	else:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (math.sin(x) * ((math.sin(x) * -0.0625) + (y * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((math.cos(x) * (-0.5 + t_0)) - t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64((sin(y) ^ 2.0) * Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)))) / Float64(Float64(3.0 + Float64(cos(x) * Float64(3.0 * Float64(Float64(sqrt(5.0) / 2.0) + -0.5)))) + Float64(1.5 * Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(sin(x) * Float64(Float64(sin(x) * -0.0625) + Float64(y * 1.00390625)))))) / Float64(3.0 + Float64(3.0 * Float64(1.5 + Float64(Float64(cos(x) * Float64(-0.5 + t_0)) - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + ((sin(y) ^ 2.0) * ((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)))) / ((3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))) + (1.5 * (cos(y) * (4.0 / (3.0 + sqrt(5.0))))));
	else
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (sin(x) * ((sin(x) * -0.0625) + (y * 1.00390625)))))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_0)) - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision] + N[(y * 1.00390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(1.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.1%

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

   5. Taylor expanded in x around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. *-commutative 61.8%

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

      6. *-commutative 61.8%

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

      7. associate-*l* 61.8%

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

   7. Simplified 61.8%

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

      1. flip-- 29.5%

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

      2. metadata-eval 29.5%

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

      3. pow1/2 29.5%

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

      4. pow1/2 29.5%

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

      5. pow-prod-up 29.5%

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

      6. metadata-eval 29.5%

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

      7. metadata-eval 29.5%

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

      8. metadata-eval 29.5%

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

   9. Applied egg-rr 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. associate--l+ 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around 0 98.6%

      \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\color{blue}{\left(-0.0625 \cdot {\sin x}^{2} + y \cdot \left(\sin x + 0.00390625 \cdot \sin x\right)\right)} \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]
   9. Step-by-step derivation

      1. unpow2 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(-0.0625 \cdot \color{blue}{\left(\sin x \cdot \sin x\right)} + y \cdot \left(\sin x + 0.00390625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      2. associate-*r* 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\color{blue}{\left(-0.0625 \cdot \sin x\right) \cdot \sin x} + y \cdot \left(\sin x + 0.00390625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      3. metadata-eval 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\color{blue}{\left(-0.0625\right)} \cdot \sin x\right) \cdot \sin x + y \cdot \left(\sin x + 0.00390625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      4. distribute-lft-neg-in 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\color{blue}{\left(-0.0625 \cdot \sin x\right)} \cdot \sin x + y \cdot \left(\sin x + 0.00390625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      5. *-commutative 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(-\color{blue}{\sin x \cdot 0.0625}\right) \cdot \sin x + y \cdot \left(\sin x + 0.00390625 \cdot \sin x\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      6. distribute-rgt1-in 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(-\sin x \cdot 0.0625\right) \cdot \sin x + y \cdot \color{blue}{\left(\left(0.00390625 + 1\right) \cdot \sin x\right)}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      7. associate-*r* 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(-\sin x \cdot 0.0625\right) \cdot \sin x + \color{blue}{\left(y \cdot \left(0.00390625 + 1\right)\right) \cdot \sin x}\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      8. distribute-rgt-out 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\color{blue}{\left(\sin x \cdot \left(\left(-\sin x \cdot 0.0625\right) + y \cdot \left(0.00390625 + 1\right)\right)\right)} \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      9. *-commutative 98.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x \cdot \left(\left(-\color{blue}{0.0625 \cdot \sin x}\right) + y \cdot \left(0.00390625 + 1\right)\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      10. distribute-lft-neg-in 98.6%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x \cdot \left(\color{blue}{\left(-0.0625\right) \cdot \sin x} + y \cdot \left(0.00390625 + 1\right)\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      11. metadata-eval 98.6%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x \cdot \left(\color{blue}{-0.0625} \cdot \sin x + y \cdot \left(0.00390625 + 1\right)\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

      12. metadata-eval 98.6%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x \cdot \left(-0.0625 \cdot \sin x + y \cdot \color{blue}{1.00390625}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]

   10. Simplified 98.6%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\color{blue}{\left(\sin x \cdot \left(-0.0625 \cdot \sin x + y \cdot 1.00390625\right)\right)} \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} + -0.5\right) - 0.5 \cdot \sqrt{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin x \cdot \left(\sin x \cdot -0.0625 + y \cdot 1.00390625\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2} + -0.5\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\cos x \cdot t\_0 + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (sqrt 5.0) 2.0) -0.5)))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+
       3.0
       (*
        3.0
        (+ (* (cos x) t_0) (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5))))))))
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
      (+
       (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))
       (+ 3.0 (* (cos x) (* 3.0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) / 2.0) + -0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((cos(x) * t_0) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	} else {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (3.0 + (cos(x) * (3.0 * t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(5.0d0) / 2.0d0) + (-0.5d0)
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (3.0d0 * ((cos(x) * t_0) + (cos(y) * (1.0d0 / (1.5d0 + (sqrt(5.0d0) * 0.5d0)))))))
    else
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / ((1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))) + (3.0d0 + (cos(x) * (3.0d0 * 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) + -0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (3.0 * ((Math.cos(x) * t_0) + (Math.cos(y) * (1.0 / (1.5 + (Math.sqrt(5.0) * 0.5)))))));
	} else {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / ((1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) + (3.0 + (Math.cos(x) * (3.0 * t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sqrt(5.0) / 2.0) + -0.5
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (3.0 * ((math.cos(x) * t_0) + (math.cos(y) * (1.0 / (1.5 + (math.sqrt(5.0) * 0.5)))))))
	else:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / ((1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))) + (3.0 + (math.cos(x) * (3.0 * t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) / 2.0) + -0.5)
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(3.0 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / Float64(Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + Float64(3.0 + Float64(cos(x) * Float64(3.0 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sqrt(5.0) / 2.0) + -0.5;
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((cos(x) * t_0) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	else
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (3.0 + (cos(x) * (3.0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.0%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow2 98.8%

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

      4. div-inv 98.8%

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

      5. metadata-eval 98.8%

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

      6. div-inv 98.8%

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

      7. metadata-eval 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

      2. +-commutative 98.8%

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

      3. neg-sub0 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{\left(0 - {\left(\sqrt{5} \cdot 0.5\right)}^{2}\right)} + 2.25}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      4. associate-+l- 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{0 - \left({\left(\sqrt{5} \cdot 0.5\right)}^{2} - 2.25\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      5. unpow2 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      6. swap-sqr 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      7. rem-square-sqrt 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right) - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      8. metadata-eval 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(5 \cdot \color{blue}{0.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      9. metadata-eval 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{1.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      10. metadata-eval 99.0%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \color{blue}{-1}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      11. metadata-eval 99.0%

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

      12. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around 0 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. sub-neg 98.4%

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

      5. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2} + -0.5\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\cos x \cdot t\_0 + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot t\_0\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (sqrt 5.0) 2.0) -0.5)))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (+
       3.0
       (*
        3.0
        (+ (* (cos x) t_0) (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5))))))))
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
      (+
       (+ 3.0 (* (cos x) (* 3.0 t_0)))
       (* 1.5 (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = (sqrt(5.0) / 2.0) + -0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((cos(x) * t_0) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	} else {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / ((3.0 + (cos(x) * (3.0 * t_0))) + (1.5 * (cos(y) * (4.0 / (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) :: tmp
    t_0 = (sqrt(5.0d0) / 2.0d0) + (-0.5d0)
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (3.0d0 * ((cos(x) * t_0) + (cos(y) * (1.0d0 / (1.5d0 + (sqrt(5.0d0) * 0.5d0)))))))
    else
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / ((3.0d0 + (cos(x) * (3.0d0 * t_0))) + (1.5d0 * (cos(y) * (4.0d0 / (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) / 2.0) + -0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 + (3.0 * ((Math.cos(x) * t_0) + (Math.cos(y) * (1.0 / (1.5 + (Math.sqrt(5.0) * 0.5)))))));
	} else {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / ((3.0 + (Math.cos(x) * (3.0 * t_0))) + (1.5 * (Math.cos(y) * (4.0 / (3.0 + Math.sqrt(5.0))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sqrt(5.0) / 2.0) + -0.5
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 + (3.0 * ((math.cos(x) * t_0) + (math.cos(y) * (1.0 / (1.5 + (math.sqrt(5.0) * 0.5)))))))
	else:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / ((3.0 + (math.cos(x) * (3.0 * t_0))) + (1.5 * (math.cos(y) * (4.0 / (3.0 + math.sqrt(5.0))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(5.0) / 2.0) + -0.5)
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 + Float64(3.0 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5))))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / Float64(Float64(3.0 + Float64(cos(x) * Float64(3.0 * t_0))) + Float64(1.5 * Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sqrt(5.0) / 2.0) + -0.5;
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * ((cos(x) * t_0) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	else
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / ((3.0 + (cos(x) * (3.0 * t_0))) + (1.5 * (cos(y) * (4.0 / (3.0 + sqrt(5.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.0%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow2 98.8%

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

      4. div-inv 98.8%

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

      5. metadata-eval 98.8%

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

      6. div-inv 98.8%

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

      7. metadata-eval 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

      2. +-commutative 98.8%

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

      3. neg-sub0 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{\left(0 - {\left(\sqrt{5} \cdot 0.5\right)}^{2}\right)} + 2.25}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      4. associate-+l- 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{0 - \left({\left(\sqrt{5} \cdot 0.5\right)}^{2} - 2.25\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      5. unpow2 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      6. swap-sqr 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      7. rem-square-sqrt 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right) - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      8. metadata-eval 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(5 \cdot \color{blue}{0.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      9. metadata-eval 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{1.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      10. metadata-eval 99.0%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \color{blue}{-1}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      11. metadata-eval 99.0%

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

      12. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around 0 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. sub-neg 98.4%

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

      5. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. pow1/2 98.3%

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

      4. pow1/2 98.3%

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

      5. pow-prod-up 98.5%

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

      6. metadata-eval 98.5%

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

      7. metadata-eval 98.5%

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

      8. metadata-eval 98.5%

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

   9. Applied egg-rr 98.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          3.0
          (*
           3.0
           (+
            (* (cos x) (+ (/ (sqrt 5.0) 2.0) -0.5))
            (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5)))))))))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      t_0)
     (/
      (+ 2.0 (* (pow (sin x) 2.0) (* (* (sqrt 2.0) -0.0625) (+ (cos x) -1.0))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = 3.0 + (3.0 * ((cos(x) * ((sqrt(5.0) / 2.0) + -0.5)) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5))))));
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + (pow(sin(x), 2.0) * ((sqrt(2.0) * -0.0625) * (cos(x) + -1.0)))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (3.0d0 * ((cos(x) * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0))) + (cos(y) * (1.0d0 / (1.5d0 + (sqrt(5.0d0) * 0.5d0))))))
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / t_0
    else
        tmp = (2.0d0 + ((sin(x) ** 2.0d0) * ((sqrt(2.0d0) * (-0.0625d0)) * (cos(x) + (-1.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 + (3.0 * ((Math.cos(x) * ((Math.sqrt(5.0) / 2.0) + -0.5)) + (Math.cos(y) * (1.0 / (1.5 + (Math.sqrt(5.0) * 0.5))))));
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / t_0;
	} else {
		tmp = (2.0 + (Math.pow(Math.sin(x), 2.0) * ((Math.sqrt(2.0) * -0.0625) * (Math.cos(x) + -1.0)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 + (3.0 * ((math.cos(x) * ((math.sqrt(5.0) / 2.0) + -0.5)) + (math.cos(y) * (1.0 / (1.5 + (math.sqrt(5.0) * 0.5))))))
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / t_0
	else:
		tmp = (2.0 + (math.pow(math.sin(x), 2.0) * ((math.sqrt(2.0) * -0.0625) * (math.cos(x) + -1.0)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 + Float64(3.0 * Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) + -0.5)) + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5)))))))
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(Float64(sqrt(2.0) * -0.0625) * Float64(cos(x) + -1.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 + (3.0 * ((cos(x) * ((sqrt(5.0) / 2.0) + -0.5)) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5))))));
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	else
		tmp = (2.0 + ((sin(x) ^ 2.0) * ((sqrt(2.0) * -0.0625) * (cos(x) + -1.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 + N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.0%

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

      1. flip-- 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow2 98.8%

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

      4. div-inv 98.8%

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

      5. metadata-eval 98.8%

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

      6. div-inv 98.8%

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

      7. metadata-eval 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. sub-neg 98.8%

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

      2. +-commutative 98.8%

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

      3. neg-sub0 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{\left(0 - {\left(\sqrt{5} \cdot 0.5\right)}^{2}\right)} + 2.25}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      4. associate-+l- 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{0 - \left({\left(\sqrt{5} \cdot 0.5\right)}^{2} - 2.25\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      5. unpow2 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      6. swap-sqr 98.8%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      7. rem-square-sqrt 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right) - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      8. metadata-eval 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(5 \cdot \color{blue}{0.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      9. metadata-eval 99.0%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{1.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      10. metadata-eval 99.0%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \color{blue}{-1}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      11. metadata-eval 99.0%

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

      12. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around 0 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow2 99.5%

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

      4. div-inv 99.5%

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

      5. metadata-eval 99.5%

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

      6. div-inv 99.5%

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

      7. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. neg-sub0 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{\left(0 - {\left(\sqrt{5} \cdot 0.5\right)}^{2}\right)} + 2.25}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      4. associate-+l- 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{0 - \left({\left(\sqrt{5} \cdot 0.5\right)}^{2} - 2.25\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      5. unpow2 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      6. swap-sqr 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      7. rem-square-sqrt 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right) - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(5 \cdot \color{blue}{0.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      9. metadata-eval 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{1.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      10. metadata-eval 99.6%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \color{blue}{-1}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 98.5%

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

       1. *-commutative 60.7%

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

       2. sub-neg 60.7%

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

       3. metadata-eval 60.7%

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

       4. *-commutative 60.7%

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

       5. associate-*l* 60.7%

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

       6. *-commutative 60.7%

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

       7. associate-*r* 60.7%

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

       8. *-commutative 60.7%

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

       9. associate-*r* 60.7%

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

   11. Simplified 98.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \frac{\sqrt{5}}{2} + -0.5\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot t\_0\right)}{\left(3 + \cos x \cdot \left(3 \cdot t\_1\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(t\_0 \cdot \left(\cos x + -1\right)\right)}{3 + 3 \cdot \left(\cos x \cdot t\_1 + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625)) (t_1 (+ (/ (sqrt 5.0) 2.0) -0.5)))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+ 2.0 (* (pow (sin y) 2.0) (* (- 1.0 (cos y)) t_0)))
      (+
       (+ 3.0 (* (cos x) (* 3.0 t_1)))
       (* 1.5 (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))))))
     (/
      (+ 2.0 (* (pow (sin x) 2.0) (* t_0 (+ (cos x) -1.0))))
      (+
       3.0
       (*
        3.0
        (+
         (* (cos x) t_1)
         (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = (sqrt(5.0) / 2.0) + -0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (pow(sin(y), 2.0) * ((1.0 - cos(y)) * t_0))) / ((3.0 + (cos(x) * (3.0 * t_1))) + (1.5 * (cos(y) * (4.0 / (3.0 + sqrt(5.0))))));
	} else {
		tmp = (2.0 + (pow(sin(x), 2.0) * (t_0 * (cos(x) + -1.0)))) / (3.0 + (3.0 * ((cos(x) * t_1) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(2.0d0) * (-0.0625d0)
    t_1 = (sqrt(5.0d0) / 2.0d0) + (-0.5d0)
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + ((sin(y) ** 2.0d0) * ((1.0d0 - cos(y)) * t_0))) / ((3.0d0 + (cos(x) * (3.0d0 * t_1))) + (1.5d0 * (cos(y) * (4.0d0 / (3.0d0 + sqrt(5.0d0))))))
    else
        tmp = (2.0d0 + ((sin(x) ** 2.0d0) * (t_0 * (cos(x) + (-1.0d0))))) / (3.0d0 + (3.0d0 * ((cos(x) * t_1) + (cos(y) * (1.0d0 / (1.5d0 + (sqrt(5.0d0) * 0.5d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = (Math.sqrt(5.0) / 2.0) + -0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (Math.pow(Math.sin(y), 2.0) * ((1.0 - Math.cos(y)) * t_0))) / ((3.0 + (Math.cos(x) * (3.0 * t_1))) + (1.5 * (Math.cos(y) * (4.0 / (3.0 + Math.sqrt(5.0))))));
	} else {
		tmp = (2.0 + (Math.pow(Math.sin(x), 2.0) * (t_0 * (Math.cos(x) + -1.0)))) / (3.0 + (3.0 * ((Math.cos(x) * t_1) + (Math.cos(y) * (1.0 / (1.5 + (Math.sqrt(5.0) * 0.5)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = (math.sqrt(5.0) / 2.0) + -0.5
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (math.pow(math.sin(y), 2.0) * ((1.0 - math.cos(y)) * t_0))) / ((3.0 + (math.cos(x) * (3.0 * t_1))) + (1.5 * (math.cos(y) * (4.0 / (3.0 + math.sqrt(5.0))))))
	else:
		tmp = (2.0 + (math.pow(math.sin(x), 2.0) * (t_0 * (math.cos(x) + -1.0)))) / (3.0 + (3.0 * ((math.cos(x) * t_1) + (math.cos(y) * (1.0 / (1.5 + (math.sqrt(5.0) * 0.5)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(Float64(sqrt(5.0) / 2.0) + -0.5)
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64((sin(y) ^ 2.0) * Float64(Float64(1.0 - cos(y)) * t_0))) / Float64(Float64(3.0 + Float64(cos(x) * Float64(3.0 * t_1))) + Float64(1.5 * Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))))));
	else
		tmp = Float64(Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(t_0 * Float64(cos(x) + -1.0)))) / Float64(3.0 + Float64(3.0 * Float64(Float64(cos(x) * t_1) + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = (sqrt(5.0) / 2.0) + -0.5;
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + ((sin(y) ^ 2.0) * ((1.0 - cos(y)) * t_0))) / ((3.0 + (cos(x) * (3.0 * t_1))) + (1.5 * (cos(y) * (4.0 / (3.0 + sqrt(5.0))))));
	else
		tmp = (2.0 + ((sin(x) ^ 2.0) * (t_0 * (cos(x) + -1.0)))) / (3.0 + (3.0 * ((cos(x) * t_1) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.1%

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

   5. Taylor expanded in x around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. *-commutative 61.8%

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

      6. *-commutative 61.8%

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

      7. associate-*l* 61.8%

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

   7. Simplified 61.8%

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

      1. flip-- 29.5%

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

      2. metadata-eval 29.5%

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

      3. pow1/2 29.5%

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

      4. pow1/2 29.5%

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

      5. pow-prod-up 29.5%

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

      6. metadata-eval 29.5%

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

      7. metadata-eval 29.5%

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

      8. metadata-eval 29.5%

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

   9. Applied egg-rr 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. flip-- 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow2 99.5%

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

      4. div-inv 99.5%

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

      5. metadata-eval 99.5%

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

      6. div-inv 99.5%

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

      7. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. neg-sub0 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{\left(0 - {\left(\sqrt{5} \cdot 0.5\right)}^{2}\right)} + 2.25}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      4. associate-+l- 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{\color{blue}{0 - \left({\left(\sqrt{5} \cdot 0.5\right)}^{2} - 2.25\right)}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      5. unpow2 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      6. swap-sqr 99.5%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      7. rem-square-sqrt 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right) - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(5 \cdot \color{blue}{0.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      9. metadata-eval 99.6%

         \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \left(\color{blue}{1.25} - 2.25\right)}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      10. metadata-eval 99.6%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\left(\sin x + \frac{\sin y}{-16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{0 - \color{blue}{-1}}{1.5 + \sqrt{5} \cdot 0.5}\right)} \]

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 98.5%

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

       1. *-commutative 60.7%

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

       2. sub-neg 60.7%

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

       3. metadata-eval 60.7%

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

       4. *-commutative 60.7%

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

       5. associate-*l* 60.7%

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

       6. *-commutative 60.7%

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

       7. associate-*r* 60.7%

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

       8. *-commutative 60.7%

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

       9. associate-*r* 60.7%

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

   11. Simplified 98.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + {\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{\left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right) + 1.5 \cdot \left(\cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)}{3 + 3 \cdot \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot y\right)}{2}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))
          (+ 3.0 (* (cos x) (* 3.0 (+ (/ (sqrt 5.0) 2.0) -0.5)))))))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+
       2.0
       (*
        (* (- 1.0 (cos y)) (* (sqrt 2.0) -0.0625))
        (- 0.5 (/ (cos (* 2.0 y)) 2.0))))
      t_0)
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
      t_0))))

C

double code(double x, double y) {
	double t_0 = (1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5))));
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)) * (0.5 - (cos((2.0 * y)) / 2.0)))) / t_0;
	} else {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))) + (3.0d0 + (cos(x) * (3.0d0 * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0)))))
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + (((1.0d0 - cos(y)) * (sqrt(2.0d0) * (-0.0625d0))) * (0.5d0 - (cos((2.0d0 * y)) / 2.0d0)))) / t_0
    else
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) + (3.0 + (Math.cos(x) * (3.0 * ((Math.sqrt(5.0) / 2.0) + -0.5))));
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (((1.0 - Math.cos(y)) * (Math.sqrt(2.0) * -0.0625)) * (0.5 - (Math.cos((2.0 * y)) / 2.0)))) / t_0;
	} else {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))) + (3.0 + (math.cos(x) * (3.0 * ((math.sqrt(5.0) / 2.0) + -0.5))))
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (((1.0 - math.cos(y)) * (math.sqrt(2.0) * -0.0625)) * (0.5 - (math.cos((2.0 * y)) / 2.0)))) / t_0
	else:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + Float64(3.0 + Float64(cos(x) * Float64(3.0 * Float64(Float64(sqrt(5.0) / 2.0) + -0.5)))))
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(1.0 - cos(y)) * Float64(sqrt(2.0) * -0.0625)) * Float64(0.5 - Float64(cos(Float64(2.0 * y)) / 2.0)))) / t_0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5))));
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + (((1.0 - cos(y)) * (sqrt(2.0) * -0.0625)) * (0.5 - (cos((2.0 * y)) / 2.0)))) / t_0;
	else
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.1%

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

   5. Taylor expanded in x around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. *-commutative 61.8%

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

      6. *-commutative 61.8%

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

      7. associate-*l* 61.8%

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

   7. Simplified 61.8%

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

      1. unpow2 23.1%

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

      2. sin-mult 23.1%

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

   9. Applied egg-rr 61.8%

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

       1. div-sub 23.1%

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

       2. +-inverses 23.1%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(y + y\right)}{2}\right)\right)\right)}{3 + 3 \cdot 1} \]

       3. cos-0 23.1%

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

       4. metadata-eval 23.1%

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

       5. count-2 23.1%

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

       6. *-commutative 23.1%

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

   11. Simplified 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. sub-neg 98.4%

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

      5. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot y\right)}{2}\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2} \cdot -0.0625\\ t_1 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot t\_0\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot y\right)}{2}\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(t\_0 \cdot \left(\cos x + -1\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + t\_1\right) - t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 2.0) -0.0625)) (t_1 (* (sqrt 5.0) 0.5)))
   (if (or (<= y -0.039) (not (<= y 6.5e-21)))
     (/
      (+ 2.0 (* (* (- 1.0 (cos y)) t_0) (- 0.5 (/ (cos (* 2.0 y)) 2.0))))
      (+
       (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))
       (+ 3.0 (* (cos x) (* 3.0 (+ (/ (sqrt 5.0) 2.0) -0.5))))))
     (/
      (+ 2.0 (* (pow (sin x) 2.0) (* t_0 (+ (cos x) -1.0))))
      (+ 3.0 (* 3.0 (+ 1.5 (- (* (cos x) (+ -0.5 t_1)) t_1))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(2.0) * -0.0625;
	double t_1 = sqrt(5.0) * 0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (((1.0 - cos(y)) * t_0) * (0.5 - (cos((2.0 * y)) / 2.0)))) / ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))));
	} else {
		tmp = (2.0 + (pow(sin(x), 2.0) * (t_0 * (cos(x) + -1.0)))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_1)) - 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(2.0d0) * (-0.0625d0)
    t_1 = sqrt(5.0d0) * 0.5d0
    if ((y <= (-0.039d0)) .or. (.not. (y <= 6.5d-21))) then
        tmp = (2.0d0 + (((1.0d0 - cos(y)) * t_0) * (0.5d0 - (cos((2.0d0 * y)) / 2.0d0)))) / ((1.5d0 * (cos(y) * (3.0d0 - sqrt(5.0d0)))) + (3.0d0 + (cos(x) * (3.0d0 * ((sqrt(5.0d0) / 2.0d0) + (-0.5d0))))))
    else
        tmp = (2.0d0 + ((sin(x) ** 2.0d0) * (t_0 * (cos(x) + (-1.0d0))))) / (3.0d0 + (3.0d0 * (1.5d0 + ((cos(x) * ((-0.5d0) + t_1)) - t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(2.0) * -0.0625;
	double t_1 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((y <= -0.039) || !(y <= 6.5e-21)) {
		tmp = (2.0 + (((1.0 - Math.cos(y)) * t_0) * (0.5 - (Math.cos((2.0 * y)) / 2.0)))) / ((1.5 * (Math.cos(y) * (3.0 - Math.sqrt(5.0)))) + (3.0 + (Math.cos(x) * (3.0 * ((Math.sqrt(5.0) / 2.0) + -0.5)))));
	} else {
		tmp = (2.0 + (Math.pow(Math.sin(x), 2.0) * (t_0 * (Math.cos(x) + -1.0)))) / (3.0 + (3.0 * (1.5 + ((Math.cos(x) * (-0.5 + t_1)) - t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(2.0) * -0.0625
	t_1 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (y <= -0.039) or not (y <= 6.5e-21):
		tmp = (2.0 + (((1.0 - math.cos(y)) * t_0) * (0.5 - (math.cos((2.0 * y)) / 2.0)))) / ((1.5 * (math.cos(y) * (3.0 - math.sqrt(5.0)))) + (3.0 + (math.cos(x) * (3.0 * ((math.sqrt(5.0) / 2.0) + -0.5)))))
	else:
		tmp = (2.0 + (math.pow(math.sin(x), 2.0) * (t_0 * (math.cos(x) + -1.0)))) / (3.0 + (3.0 * (1.5 + ((math.cos(x) * (-0.5 + t_1)) - t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(2.0) * -0.0625)
	t_1 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((y <= -0.039) || !(y <= 6.5e-21))
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(1.0 - cos(y)) * t_0) * Float64(0.5 - Float64(cos(Float64(2.0 * y)) / 2.0)))) / Float64(Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + Float64(3.0 + Float64(cos(x) * Float64(3.0 * Float64(Float64(sqrt(5.0) / 2.0) + -0.5))))));
	else
		tmp = Float64(Float64(2.0 + Float64((sin(x) ^ 2.0) * Float64(t_0 * Float64(cos(x) + -1.0)))) / Float64(3.0 + Float64(3.0 * Float64(1.5 + Float64(Float64(cos(x) * Float64(-0.5 + t_1)) - t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(2.0) * -0.0625;
	t_1 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((y <= -0.039) || ~((y <= 6.5e-21)))
		tmp = (2.0 + (((1.0 - cos(y)) * t_0) * (0.5 - (cos((2.0 * y)) / 2.0)))) / ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (3.0 + (cos(x) * (3.0 * ((sqrt(5.0) / 2.0) + -0.5)))));
	else
		tmp = (2.0 + ((sin(x) ^ 2.0) * (t_0 * (cos(x) + -1.0)))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_1)) - t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[y, -0.039], N[Not[LessEqual[y, 6.5e-21]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(0.5 - N[(N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[Cos[x], $MachinePrecision] * N[(3.0 * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(1.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0389999999999999999 or 6.49999999999999987e-21 < 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 99.1%

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

   5. Taylor expanded in x around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.8%

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

      3. associate-*r* 61.8%

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

      4. *-commutative 61.8%

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

      5. *-commutative 61.8%

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

      6. *-commutative 61.8%

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

      7. associate-*l* 61.8%

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

   7. Simplified 61.8%

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

      1. unpow2 23.1%

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

      2. sin-mult 23.1%

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

   9. Applied egg-rr 61.8%

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

       1. div-sub 23.1%

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

       2. +-inverses 23.1%

          \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(y + y\right)}{2}\right)\right)\right)}{3 + 3 \cdot 1} \]

       3. cos-0 23.1%

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

       4. metadata-eval 23.1%

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

       5. count-2 23.1%

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

       6. *-commutative 23.1%

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

   11. Simplified 61.8%

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

   if -0.0389999999999999999 < y < 6.49999999999999987e-21

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. associate--l+ 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around 0 98.4%

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

      1. *-commutative 60.7%

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

      2. sub-neg 60.7%

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

      3. metadata-eval 60.7%

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

      4. *-commutative 60.7%

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

      5. associate-*l* 60.7%

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

      6. *-commutative 60.7%

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

      7. associate-*r* 60.7%

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

      8. *-commutative 60.7%

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

      9. associate-*r* 60.7%

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

   10. Simplified 98.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.039 \lor \neg \left(y \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{2 + \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot -0.0625\right)\right) \cdot \left(0.5 - \frac{\cos \left(2 \cdot y\right)}{2}\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + \left(3 + \cos x \cdot \left(3 \cdot \left(\frac{\sqrt{5}}{2} + -0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + {\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + \sqrt{5} \cdot 0.5\right) - \sqrt{5} \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + t\_0\right) - t\_0\right)\right)} \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	return (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_0)) - 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 = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 + (3.0d0 * (1.5d0 + ((cos(x) * ((-0.5d0) + t_0)) - t_0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 59.6%

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

   1. associate--l+ 59.6%

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

   2. sub-neg 59.6%

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

   3. metadata-eval 59.6%

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

7. Simplified 59.6%

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

8. Taylor expanded in x around 0 43.1%

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

9. Final simplification 43.1%

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

Alternative 21: 59.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{2 + {\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)}{3 + 3 \cdot \left(1.5 + \left(\cos x \cdot \left(-0.5 + t\_0\right) - t\_0\right)\right)} \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	return (2.0 + (pow(sin(x), 2.0) * ((sqrt(2.0) * -0.0625) * (cos(x) + -1.0)))) / (3.0 + (3.0 * (1.5 + ((cos(x) * (-0.5 + t_0)) - 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 = (2.0d0 + ((sin(x) ** 2.0d0) * ((sqrt(2.0d0) * (-0.0625d0)) * (cos(x) + (-1.0d0))))) / (3.0d0 + (3.0d0 * (1.5d0 + ((cos(x) * ((-0.5d0) + t_0)) - t_0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 59.6%

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

   1. associate--l+ 59.6%

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

   2. sub-neg 59.6%

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

   3. metadata-eval 59.6%

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

7. Simplified 59.6%

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

8. Taylor expanded in y around 0 59.2%

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

   1. *-commutative 40.9%

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

   2. sub-neg 40.9%

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

   3. metadata-eval 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-*l* 40.9%

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

   6. *-commutative 40.9%

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

   7. associate-*r* 40.9%

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

   8. *-commutative 40.9%

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

   9. associate-*r* 40.9%

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

10. Simplified 59.2%

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

11. Final simplification 59.2%

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

Alternative 22: 40.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \sqrt{0.0078125 \cdot \left({\left(1 - \cos y\right)}^{2} \cdot {\sin y}^{4}\right)}}{6} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+ 2.0 (sqrt (* 0.0078125 (* (pow (- 1.0 (cos y)) 2.0) (pow (sin y) 4.0)))))
  6.0))

C

double code(double x, double y) {
	return (2.0 + sqrt((0.0078125 * (pow((1.0 - cos(y)), 2.0) * pow(sin(y), 4.0))))) / 6.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + sqrt((0.0078125d0 * (((1.0d0 - cos(y)) ** 2.0d0) * (sin(y) ** 4.0d0))))) / 6.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 + Math.sqrt((0.0078125 * (Math.pow((1.0 - Math.cos(y)), 2.0) * Math.pow(Math.sin(y), 4.0))))) / 6.0;
}

Python

def code(x, y):
	return (2.0 + math.sqrt((0.0078125 * (math.pow((1.0 - math.cos(y)), 2.0) * math.pow(math.sin(y), 4.0))))) / 6.0

Julia

function code(x, y)
	return Float64(Float64(2.0 + sqrt(Float64(0.0078125 * Float64((Float64(1.0 - cos(y)) ^ 2.0) * (sin(y) ^ 4.0))))) / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + sqrt((0.0078125 * (((1.0 - cos(y)) ^ 2.0) * (sin(y) ^ 4.0))))) / 6.0;
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[Sqrt[N[(0.0078125 * N[(N[Power[N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 59.6%

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

   1. associate--l+ 59.6%

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

   2. sub-neg 59.6%

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

   3. metadata-eval 59.6%

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

7. Simplified 59.6%

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

8. Taylor expanded in x around 0 40.9%

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

9. Taylor expanded in x around 0 40.8%

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

    1. associate-*r* 40.8%

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

    2. *-commutative 40.8%

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

11. Simplified 40.8%

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

    1. add-sqr-sqrt 29.6%

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

    2. sqrt-unprod 41.0%

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

    3. pow2 41.0%

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

    4. associate-*r* 41.0%

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

    5. *-commutative 41.0%

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

13. Applied egg-rr 41.0%

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

    1. unpow2 41.0%

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

    2. associate-*l* 41.0%

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

    3. associate-*r* 41.0%

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

    4. *-commutative 41.0%

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

    5. *-commutative 41.0%

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

    6. associate-*r* 41.0%

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

    7. associate-*l* 41.0%

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

    8. associate-*r* 41.0%

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

    9. *-commutative 41.0%

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

    10. *-commutative 41.0%

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

    11. associate-*r* 41.0%

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

15. Simplified 41.0%

    \[\leadsto \frac{2 + \color{blue}{\sqrt{0.0078125 \cdot \left({\left(1 - \cos y\right)}^{2} \cdot {\sin y}^{4}\right)}}}{3 + 3 \cdot 1} \]

16. Final simplification 41.0%

    \[\leadsto \frac{2 + \sqrt{0.0078125 \cdot \left({\left(1 - \cos y\right)}^{2} \cdot {\sin y}^{4}\right)}}{6} \]
17. Add Preprocessing

Alternative 23: 40.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((sin(x) ** 2.0d0) * ((sqrt(2.0d0) * (-0.0625d0)) * (cos(x) + (-1.0d0))))) / 6.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 59.6%

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

   1. associate--l+ 59.6%

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

   2. sub-neg 59.6%

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

   3. metadata-eval 59.6%

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

7. Simplified 59.6%

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

8. Taylor expanded in x around 0 40.9%

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

9. Taylor expanded in y around 0 40.9%

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

    1. *-commutative 40.9%

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

    2. sub-neg 40.9%

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

    3. metadata-eval 40.9%

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

    4. *-commutative 40.9%

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

    5. associate-*l* 40.9%

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

    6. *-commutative 40.9%

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

    7. associate-*r* 40.9%

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

    8. *-commutative 40.9%

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

    9. associate-*r* 40.9%

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

11. Simplified 40.9%

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

12. Final simplification 40.9%

    \[\leadsto \frac{2 + {\sin x}^{2} \cdot \left(\left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\cos x + -1\right)\right)}{6} \]
13. Add Preprocessing

Alternative 24: 40.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(0.5 - \frac{\cos \left(2 \cdot y\right)}{2}\right)\right)\right)}{6} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (sqrt 2.0)
    (* (- 1.0 (cos y)) (* -0.0625 (- 0.5 (/ (cos (* 2.0 y)) 2.0))))))
  6.0))

C

double code(double x, double y) {
	return (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * (-0.0625 * (0.5 - (cos((2.0 * y)) / 2.0)))))) / 6.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((1.0d0 - cos(y)) * ((-0.0625d0) * (0.5d0 - (cos((2.0d0 * y)) / 2.0d0)))))) / 6.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 + (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * (-0.0625 * (0.5 - (Math.cos((2.0 * y)) / 2.0)))))) / 6.0;
}

Python

def code(x, y):
	return (2.0 + (math.sqrt(2.0) * ((1.0 - math.cos(y)) * (-0.0625 * (0.5 - (math.cos((2.0 * y)) / 2.0)))))) / 6.0

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(-0.0625 * Float64(0.5 - Float64(cos(Float64(2.0 * y)) / 2.0)))))) / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + (sqrt(2.0) * ((1.0 - cos(y)) * (-0.0625 * (0.5 - (cos((2.0 * y)) / 2.0)))))) / 6.0;
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(0.5 - N[(N[Cos[N[(2.0 * y), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 59.6%

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

   1. associate--l+ 59.6%

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

   2. sub-neg 59.6%

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

   3. metadata-eval 59.6%

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

7. Simplified 59.6%

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

8. Taylor expanded in x around 0 40.9%

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

9. Taylor expanded in x around 0 40.8%

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

    1. associate-*r* 40.8%

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

    2. *-commutative 40.8%

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

11. Simplified 40.8%

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

    1. unpow2 40.8%

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

    2. sin-mult 40.8%

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

13. Applied egg-rr 40.8%

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

    1. div-sub 40.8%

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

    2. +-inverses 40.8%

       \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(y + y\right)}{2}\right)\right)\right)}{3 + 3 \cdot 1} \]

    3. cos-0 40.8%

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

    4. metadata-eval 40.8%

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

    5. count-2 40.8%

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

    6. *-commutative 40.8%

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

15. Simplified 40.8%

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

16. Final simplification 40.8%

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot \left(0.5 - \frac{\cos \left(2 \cdot y\right)}{2}\right)\right)\right)}{6} \]
17. Add Preprocessing

Alternative 25: 30.4% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (+ 2.0 (* (sqrt 2.0) (* (- 1.0 (cos y)) (* -0.0625 (pow y 2.0))))) 6.0))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + (sqrt(2.0d0) * ((1.0d0 - cos(y)) * ((-0.0625d0) * (y ** 2.0d0))))) / 6.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 59.6%

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

   1. associate--l+ 59.6%

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

   2. sub-neg 59.6%

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

   3. metadata-eval 59.6%

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

7. Simplified 59.6%

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

8. Taylor expanded in x around 0 40.9%

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

9. Taylor expanded in x around 0 40.8%

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

    1. associate-*r* 40.8%

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

    2. *-commutative 40.8%

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

11. Simplified 40.8%

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

12. Taylor expanded in y around 0 30.8%

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

13. Final simplification 30.8%

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(-0.0625 \cdot {y}^{2}\right)\right)}{6} \]
14. Add Preprocessing

Alternative 26: 30.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{6} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (+ 2.0 (* (pow y 4.0) (* (sqrt 2.0) -0.03125))) 6.0))

C

double code(double x, double y) {
	return (2.0 + (pow(y, 4.0) * (sqrt(2.0) * -0.03125))) / 6.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((y ** 4.0d0) * (sqrt(2.0d0) * (-0.03125d0)))) / 6.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 + (Math.pow(y, 4.0) * (Math.sqrt(2.0) * -0.03125))) / 6.0;
}

Python

def code(x, y):
	return (2.0 + (math.pow(y, 4.0) * (math.sqrt(2.0) * -0.03125))) / 6.0

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64((y ^ 4.0) * Float64(sqrt(2.0) * -0.03125))) / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((y ^ 4.0) * (sqrt(2.0) * -0.03125))) / 6.0;
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[Power[y, 4.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.03125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.2%

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

5. Taylor expanded in y around 0 59.6%

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

   1. associate--l+ 59.6%

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

   2. sub-neg 59.6%

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

   3. metadata-eval 59.6%

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

7. Simplified 59.6%

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

8. Taylor expanded in x around 0 40.9%

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

9. Taylor expanded in x around 0 40.8%

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

    1. associate-*r* 40.8%

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

    2. *-commutative 40.8%

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

11. Simplified 40.8%

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

12. Taylor expanded in y around 0 30.7%

    \[\leadsto \frac{2 + \color{blue}{-0.03125 \cdot \left({y}^{4} \cdot \sqrt{2}\right)}}{3 + 3 \cdot 1} \]
13. Step-by-step derivation

    1. *-commutative 30.7%

       \[\leadsto \frac{2 + \color{blue}{\left({y}^{4} \cdot \sqrt{2}\right) \cdot -0.03125}}{3 + 3 \cdot 1} \]

    2. associate-*l* 30.7%

       \[\leadsto \frac{2 + \color{blue}{{y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}}{3 + 3 \cdot 1} \]

14. Simplified 30.7%

    \[\leadsto \frac{2 + \color{blue}{{y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}}{3 + 3 \cdot 1} \]

15. Final simplification 30.7%

    \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{6} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024048 
(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))))))