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

Percentage Accurate: 99.3% → 99.3%

Time: 47.0s

Alternatives: 29

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around inf 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in x around inf 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. sub-neg 99.3%

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

   4. *-commutative 99.3%

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

   5. metadata-eval 99.3%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. *-commutative 99.3%

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

   9. fma-def 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around -inf 99.3%

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

3. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around inf 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*r* 99.3%

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

   5. *-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. associate-*l* 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around inf 99.3%

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

4. Applied egg-rr 99.3%

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

5. Final simplification 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around -inf 99.3%

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

3. Taylor expanded in x around inf 99.2%

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

   1. distribute-lft-in 65.2%

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

   2. metadata-eval 65.2%

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

   3. distribute-lft-out 65.2%

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

   4. *-commutative 65.2%

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

   5. sub-neg 65.2%

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

   6. metadata-eval 65.2%

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

   7. *-commutative 65.2%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 6: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot t_0\right)\\ t_2 := \frac{\sqrt{5}}{2}\\ t_3 := 1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\\ \mathbf{if}\;x \leq -0.0285:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + t_3\right)}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(t_0 \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_2 - 0.5\right) + \cos y \cdot \left(1.5 - t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(t_3 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1 (+ 2.0 (* (- (cos x) (cos y)) (* (* (sqrt 2.0) (sin x)) t_0))))
        (t_2 (/ (sqrt 5.0) 2.0))
        (t_3 (+ 1.0 (/ (+ (sqrt 5.0) -1.0) (/ 2.0 (cos x))))))
   (if (<= x -0.0285)
     (/ t_1 (* 3.0 (+ (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)) t_3)))
     (if (<= x 0.8)
       (/
        (+
         2.0
         (*
          (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))
          (* t_0 (+ 1.0 (- (* -0.5 (* x x)) (cos y))))))
        (* 3.0 (+ 1.0 (+ (* (cos x) (- t_2 0.5)) (* (cos y) (- 1.5 t_2))))))
       (/
        t_1
        (* 3.0 (+ t_3 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0));
	double t_2 = sqrt(5.0) / 2.0;
	double t_3 = 1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x)));
	double tmp;
	if (x <= -0.0285) {
		tmp = t_1 / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + t_3));
	} else if (x <= 0.8) {
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - cos(y)))))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	} else {
		tmp = t_1 / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * t_0))
    t_2 = sqrt(5.0d0) / 2.0d0
    t_3 = 1.0d0 + ((sqrt(5.0d0) + (-1.0d0)) / (2.0d0 / cos(x)))
    if (x <= (-0.0285d0)) then
        tmp = t_1 / (3.0d0 * ((cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)) + t_3))
    else if (x <= 0.8d0) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))) * (t_0 * (1.0d0 + (((-0.5d0) * (x * x)) - cos(y)))))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_2 - 0.5d0)) + (cos(y) * (1.5d0 - t_2)))))
    else
        tmp = t_1 / (3.0d0 * (t_3 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * t_0));
	double t_2 = Math.sqrt(5.0) / 2.0;
	double t_3 = 1.0 + ((Math.sqrt(5.0) + -1.0) / (2.0 / Math.cos(x)));
	double tmp;
	if (x <= -0.0285) {
		tmp = t_1 / (3.0 * ((Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)) + t_3));
	} else if (x <= 0.8) {
		tmp = (2.0 + ((Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - Math.cos(y)))))) / (3.0 * (1.0 + ((Math.cos(x) * (t_2 - 0.5)) + (Math.cos(y) * (1.5 - t_2)))));
	} else {
		tmp = t_1 / (3.0 * (t_3 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * t_0))
	t_2 = math.sqrt(5.0) / 2.0
	t_3 = 1.0 + ((math.sqrt(5.0) + -1.0) / (2.0 / math.cos(x)))
	tmp = 0
	if x <= -0.0285:
		tmp = t_1 / (3.0 * ((math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)) + t_3))
	elif x <= 0.8:
		tmp = (2.0 + ((math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - math.cos(y)))))) / (3.0 * (1.0 + ((math.cos(x) * (t_2 - 0.5)) + (math.cos(y) * (1.5 - t_2)))))
	else:
		tmp = t_1 / (3.0 * (t_3 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * t_0)))
	t_2 = Float64(sqrt(5.0) / 2.0)
	t_3 = Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) / Float64(2.0 / cos(x))))
	tmp = 0.0
	if (x <= -0.0285)
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)) + t_3)));
	elseif (x <= 0.8)
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))) * Float64(t_0 * Float64(1.0 + Float64(Float64(-0.5 * Float64(x * x)) - cos(y)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_2 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_2))))));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * t_0));
	t_2 = sqrt(5.0) / 2.0;
	t_3 = 1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x)));
	tmp = 0.0;
	if (x <= -0.0285)
		tmp = t_1 / (3.0 * ((cos(y) * ((3.0 - sqrt(5.0)) / 2.0)) + t_3));
	elseif (x <= 0.8)
		tmp = (2.0 + ((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (t_0 * (1.0 + ((-0.5 * (x * x)) - cos(y)))))) / (3.0 * (1.0 + ((cos(x) * (t_2 - 0.5)) + (cos(y) * (1.5 - t_2)))));
	else
		tmp = t_1 / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(2.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0285], N[(t$95$1 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.8], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 + N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.028500000000000001

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.3%

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

      1. associate-*l/ 61.3%

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

      2. sub-neg 61.3%

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

      3. metadata-eval 61.3%

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

   4. Applied egg-rr 61.3%

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

      1. metadata-eval 61.3%

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

      2. sub-neg 61.3%

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

      3. associate-/l* 61.3%

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

      4. sub-neg 61.3%

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

      5. metadata-eval 61.3%

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

   6. Simplified 61.3%

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

   if -0.028500000000000001 < x < 0.80000000000000004

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. associate-+l+ 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. div-sub 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.0%

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

      1. associate--l+ 99.0%

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

      2. unpow2 99.0%

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

   6. Simplified 99.0%

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

   if 0.80000000000000004 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 61.7%

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

      1. associate-*l/ 61.7%

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

      2. sub-neg 61.7%

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

      3. metadata-eval 61.7%

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

   4. Applied egg-rr 61.7%

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

      1. metadata-eval 61.7%

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

      2. sub-neg 61.7%

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

      3. associate-/l* 61.7%

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

      4. sub-neg 61.7%

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

      5. metadata-eval 61.7%

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

   6. Simplified 61.7%

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

      1. flip-- 26.0%

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

      2. metadata-eval 26.0%

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

      3. add-sqr-sqrt 26.0%

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

      4. metadata-eval 26.0%

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

   8. Applied egg-rr 61.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0285:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 + \left(-0.5 \cdot \left(x \cdot x\right) - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 7: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_1 := 3 - \sqrt{5}\\ t_2 := 1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\\ \mathbf{if}\;x \leq -0.048:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(\cos y \cdot \frac{t_1}{2} + t_2\right)}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\frac{\frac{2 + \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sqrt{2} \cdot \left(-0.5 \cdot \left(x \cdot x\right) + \left(1 - \cos y\right)\right)\right)}{3}}{1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(0.5 \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{3 \cdot \left(t_2 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (+
          2.0
          (*
           (- (cos x) (cos y))
           (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))))))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (+ 1.0 (/ (+ (sqrt 5.0) -1.0) (/ 2.0 (cos x))))))
   (if (<= x -0.048)
     (/ t_0 (* 3.0 (+ (* (cos y) (/ t_1 2.0)) t_2)))
     (if (<= x 0.8)
       (/
        (/
         (+
          2.0
          (*
           (* (+ (sin x) (* -0.0625 (sin y))) (+ (sin y) (* (sin x) -0.0625)))
           (* (sqrt 2.0) (+ (* -0.5 (* x x)) (- 1.0 (cos y))))))
         3.0)
        (+
         1.0
         (+ (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5)) (* (cos y) (* 0.5 t_1)))))
       (/
        t_0
        (* 3.0 (+ t_2 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))));
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = 1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x)));
	double tmp;
	if (x <= -0.048) {
		tmp = t_0 / (3.0 * ((cos(y) * (t_1 / 2.0)) + t_2));
	} else if (x <= 0.8) {
		tmp = ((2.0 + (((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))) * (sqrt(2.0) * ((-0.5 * (x * x)) + (1.0 - cos(y)))))) / 3.0) / (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (0.5 * t_1))));
	} else {
		tmp = t_0 / (3.0 * (t_2 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))
    t_1 = 3.0d0 - sqrt(5.0d0)
    t_2 = 1.0d0 + ((sqrt(5.0d0) + (-1.0d0)) / (2.0d0 / cos(x)))
    if (x <= (-0.048d0)) then
        tmp = t_0 / (3.0d0 * ((cos(y) * (t_1 / 2.0d0)) + t_2))
    else if (x <= 0.8d0) then
        tmp = ((2.0d0 + (((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) + (sin(x) * (-0.0625d0)))) * (sqrt(2.0d0) * (((-0.5d0) * (x * x)) + (1.0d0 - cos(y)))))) / 3.0d0) / (1.0d0 + ((cos(x) * ((sqrt(5.0d0) / 2.0d0) - 0.5d0)) + (cos(y) * (0.5d0 * t_1))))
    else
        tmp = t_0 / (3.0d0 * (t_2 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))));
	double t_1 = 3.0 - Math.sqrt(5.0);
	double t_2 = 1.0 + ((Math.sqrt(5.0) + -1.0) / (2.0 / Math.cos(x)));
	double tmp;
	if (x <= -0.048) {
		tmp = t_0 / (3.0 * ((Math.cos(y) * (t_1 / 2.0)) + t_2));
	} else if (x <= 0.8) {
		tmp = ((2.0 + (((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) + (Math.sin(x) * -0.0625))) * (Math.sqrt(2.0) * ((-0.5 * (x * x)) + (1.0 - Math.cos(y)))))) / 3.0) / (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) / 2.0) - 0.5)) + (Math.cos(y) * (0.5 * t_1))));
	} else {
		tmp = t_0 / (3.0 * (t_2 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))
	t_1 = 3.0 - math.sqrt(5.0)
	t_2 = 1.0 + ((math.sqrt(5.0) + -1.0) / (2.0 / math.cos(x)))
	tmp = 0
	if x <= -0.048:
		tmp = t_0 / (3.0 * ((math.cos(y) * (t_1 / 2.0)) + t_2))
	elif x <= 0.8:
		tmp = ((2.0 + (((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) + (math.sin(x) * -0.0625))) * (math.sqrt(2.0) * ((-0.5 * (x * x)) + (1.0 - math.cos(y)))))) / 3.0) / (1.0 + ((math.cos(x) * ((math.sqrt(5.0) / 2.0) - 0.5)) + (math.cos(y) * (0.5 * t_1))))
	else:
		tmp = t_0 / (3.0 * (t_2 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) / Float64(2.0 / cos(x))))
	tmp = 0.0
	if (x <= -0.048)
		tmp = Float64(t_0 / Float64(3.0 * Float64(Float64(cos(y) * Float64(t_1 / 2.0)) + t_2)));
	elseif (x <= 0.8)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) + Float64(sin(x) * -0.0625))) * Float64(sqrt(2.0) * Float64(Float64(-0.5 * Float64(x * x)) + Float64(1.0 - cos(y)))))) / 3.0) / Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5)) + Float64(cos(y) * Float64(0.5 * t_1)))));
	else
		tmp = Float64(t_0 / Float64(3.0 * Float64(t_2 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))));
	t_1 = 3.0 - sqrt(5.0);
	t_2 = 1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x)));
	tmp = 0.0;
	if (x <= -0.048)
		tmp = t_0 / (3.0 * ((cos(y) * (t_1 / 2.0)) + t_2));
	elseif (x <= 0.8)
		tmp = ((2.0 + (((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))) * (sqrt(2.0) * ((-0.5 * (x * x)) + (1.0 - cos(y)))))) / 3.0) / (1.0 + ((cos(x) * ((sqrt(5.0) / 2.0) - 0.5)) + (cos(y) * (0.5 * t_1))));
	else
		tmp = t_0 / (3.0 * (t_2 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(2.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.048], N[(t$95$0 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.8], N[(N[(N[(2.0 + N[(N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(3.0 * N[(t$95$2 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.048000000000000001

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.3%

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

      1. associate-*l/ 61.3%

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

      2. sub-neg 61.3%

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

      3. metadata-eval 61.3%

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

   4. Applied egg-rr 61.3%

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

      1. metadata-eval 61.3%

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

      2. sub-neg 61.3%

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

      3. associate-/l* 61.3%

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

      4. sub-neg 61.3%

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

      5. metadata-eval 61.3%

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

   6. Simplified 61.3%

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

   if -0.048000000000000001 < x < 0.80000000000000004

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

      3. *-commutative 99.2%

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

      4. distribute-lft-out 99.2%

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

      5. unpow2 99.2%

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

   7. Simplified 99.2%

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

   if 0.80000000000000004 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 61.7%

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

      1. associate-*l/ 61.7%

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

      2. sub-neg 61.7%

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

      3. metadata-eval 61.7%

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

   4. Applied egg-rr 61.7%

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

      1. metadata-eval 61.7%

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

      2. sub-neg 61.7%

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

      3. associate-/l* 61.7%

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

      4. sub-neg 61.7%

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

      5. metadata-eval 61.7%

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

   6. Simplified 61.7%

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

      1. flip-- 26.0%

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

      2. metadata-eval 26.0%

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

      3. add-sqr-sqrt 26.0%

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

      4. metadata-eval 26.0%

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

   8. Applied egg-rr 61.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.048:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\frac{\frac{2 + \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right) \cdot \left(\sqrt{2} \cdot \left(-0.5 \cdot \left(x \cdot x\right) + \left(1 - \cos y\right)\right)\right)}{3}}{1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 8: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \cos x - \cos y\\ t_2 := 2 + t_1 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_3 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(\cos y \cdot \frac{t_3}{2} + \left(1 + \frac{t_0}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00027:\\ \;\;\;\;\frac{\frac{2 + \left(\sqrt{2} \cdot t_1\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3}}{1 + \left(-0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- (cos x) (cos y)))
        (t_2
         (+
          2.0
          (* t_1 (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))))))
        (t_3 (- 3.0 (sqrt 5.0))))
   (if (<= x -3.5e-7)
     (/
      t_2
      (* 3.0 (+ (* (cos y) (/ t_3 2.0)) (+ 1.0 (/ t_0 (/ 2.0 (cos x)))))))
     (if (<= x 0.00027)
       (/
        (/
         (+
          2.0
          (*
           (* (sqrt 2.0) t_1)
           (*
            (+ (sin x) (* -0.0625 (sin y)))
            (+ (sin y) (* (sin x) -0.0625)))))
         3.0)
        (+ 1.0 (+ -0.5 (* 0.5 (+ (sqrt 5.0) (* (cos y) t_3))))))
       (/
        t_2
        (*
         3.0
         (+
          (+ 1.0 (* (cos x) (/ t_0 2.0)))
          (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = cos(x) - cos(y);
	double t_2 = 2.0 + (t_1 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))));
	double t_3 = 3.0 - sqrt(5.0);
	double tmp;
	if (x <= -3.5e-7) {
		tmp = t_2 / (3.0 * ((cos(y) * (t_3 / 2.0)) + (1.0 + (t_0 / (2.0 / cos(x))))));
	} else if (x <= 0.00027) {
		tmp = ((2.0 + ((sqrt(2.0) * t_1) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (sqrt(5.0) + (cos(y) * t_3)))));
	} else {
		tmp = t_2 / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = cos(x) - cos(y)
    t_2 = 2.0d0 + (t_1 * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))
    t_3 = 3.0d0 - sqrt(5.0d0)
    if (x <= (-3.5d-7)) then
        tmp = t_2 / (3.0d0 * ((cos(y) * (t_3 / 2.0d0)) + (1.0d0 + (t_0 / (2.0d0 / cos(x))))))
    else if (x <= 0.00027d0) then
        tmp = ((2.0d0 + ((sqrt(2.0d0) * t_1) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) + (sin(x) * (-0.0625d0)))))) / 3.0d0) / (1.0d0 + ((-0.5d0) + (0.5d0 * (sqrt(5.0d0) + (cos(y) * t_3)))))
    else
        tmp = t_2 / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = Math.cos(x) - Math.cos(y);
	double t_2 = 2.0 + (t_1 * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))));
	double t_3 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if (x <= -3.5e-7) {
		tmp = t_2 / (3.0 * ((Math.cos(y) * (t_3 / 2.0)) + (1.0 + (t_0 / (2.0 / Math.cos(x))))));
	} else if (x <= 0.00027) {
		tmp = ((2.0 + ((Math.sqrt(2.0) * t_1) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) + (Math.sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (Math.sqrt(5.0) + (Math.cos(y) * t_3)))));
	} else {
		tmp = t_2 / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = math.cos(x) - math.cos(y)
	t_2 = 2.0 + (t_1 * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))
	t_3 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if x <= -3.5e-7:
		tmp = t_2 / (3.0 * ((math.cos(y) * (t_3 / 2.0)) + (1.0 + (t_0 / (2.0 / math.cos(x))))))
	elif x <= 0.00027:
		tmp = ((2.0 + ((math.sqrt(2.0) * t_1) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) + (math.sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (math.sqrt(5.0) + (math.cos(y) * t_3)))))
	else:
		tmp = t_2 / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(cos(x) - cos(y))
	t_2 = Float64(2.0 + Float64(t_1 * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_3 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if (x <= -3.5e-7)
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(cos(y) * Float64(t_3 / 2.0)) + Float64(1.0 + Float64(t_0 / Float64(2.0 / cos(x)))))));
	elseif (x <= 0.00027)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * t_1) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) + Float64(sin(x) * -0.0625))))) / 3.0) / Float64(1.0 + Float64(-0.5 + Float64(0.5 * Float64(sqrt(5.0) + Float64(cos(y) * t_3))))));
	else
		tmp = Float64(t_2 / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = cos(x) - cos(y);
	t_2 = 2.0 + (t_1 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))));
	t_3 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if (x <= -3.5e-7)
		tmp = t_2 / (3.0 * ((cos(y) * (t_3 / 2.0)) + (1.0 + (t_0 / (2.0 / cos(x))))));
	elseif (x <= 0.00027)
		tmp = ((2.0 + ((sqrt(2.0) * t_1) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (sqrt(5.0) + (cos(y) * t_3)))));
	else
		tmp = t_2 / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e-7], N[(t$95$2 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$3 / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(t$95$0 / N[(2.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00027], N[(N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / N[(1.0 + N[(-0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999984e-7

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.9%

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

      1. associate-*l/ 61.9%

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

      2. sub-neg 61.9%

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

      3. metadata-eval 61.9%

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

   4. Applied egg-rr 61.9%

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

      1. metadata-eval 61.9%

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

      2. sub-neg 61.9%

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

      3. associate-/l* 61.9%

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

      4. sub-neg 61.9%

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

      5. metadata-eval 61.9%

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

   6. Simplified 61.9%

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

   if -3.49999999999999984e-7 < x < 2.70000000000000003e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-lft-out 99.4%

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

      3. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if 2.70000000000000003e-4 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 61.2%

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

      1. flip-- 25.9%

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

      2. metadata-eval 25.9%

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

      3. add-sqr-sqrt 25.9%

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

      4. metadata-eval 25.9%

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

   4. Applied egg-rr 61.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00027:\\ \;\;\;\;\frac{\frac{2 + \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3}}{1 + \left(-0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 9: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 2 + t_0 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\\ t_2 := 3 - \sqrt{5}\\ t_3 := 1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(\cos y \cdot \frac{t_2}{2} + t_3\right)}\\ \mathbf{elif}\;x \leq 0.00029:\\ \;\;\;\;\frac{\frac{2 + \left(\sqrt{2} \cdot t_0\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3}}{1 + \left(-0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{3 \cdot \left(t_3 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1
         (+
          2.0
          (* t_0 (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0))))))
        (t_2 (- 3.0 (sqrt 5.0)))
        (t_3 (+ 1.0 (/ (+ (sqrt 5.0) -1.0) (/ 2.0 (cos x))))))
   (if (<= x -3.5e-7)
     (/ t_1 (* 3.0 (+ (* (cos y) (/ t_2 2.0)) t_3)))
     (if (<= x 0.00029)
       (/
        (/
         (+
          2.0
          (*
           (* (sqrt 2.0) t_0)
           (*
            (+ (sin x) (* -0.0625 (sin y)))
            (+ (sin y) (* (sin x) -0.0625)))))
         3.0)
        (+ 1.0 (+ -0.5 (* 0.5 (+ (sqrt 5.0) (* (cos y) t_2))))))
       (/
        t_1
        (* 3.0 (+ t_3 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 2.0 + (t_0 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))));
	double t_2 = 3.0 - sqrt(5.0);
	double t_3 = 1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x)));
	double tmp;
	if (x <= -3.5e-7) {
		tmp = t_1 / (3.0 * ((cos(y) * (t_2 / 2.0)) + t_3));
	} else if (x <= 0.00029) {
		tmp = ((2.0 + ((sqrt(2.0) * t_0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (sqrt(5.0) + (cos(y) * t_2)))));
	} else {
		tmp = t_1 / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = cos(x) - cos(y)
    t_1 = 2.0d0 + (t_0 * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))
    t_2 = 3.0d0 - sqrt(5.0d0)
    t_3 = 1.0d0 + ((sqrt(5.0d0) + (-1.0d0)) / (2.0d0 / cos(x)))
    if (x <= (-3.5d-7)) then
        tmp = t_1 / (3.0d0 * ((cos(y) * (t_2 / 2.0d0)) + t_3))
    else if (x <= 0.00029d0) then
        tmp = ((2.0d0 + ((sqrt(2.0d0) * t_0) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) + (sin(x) * (-0.0625d0)))))) / 3.0d0) / (1.0d0 + ((-0.5d0) + (0.5d0 * (sqrt(5.0d0) + (cos(y) * t_2)))))
    else
        tmp = t_1 / (3.0d0 * (t_3 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = 2.0 + (t_0 * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))));
	double t_2 = 3.0 - Math.sqrt(5.0);
	double t_3 = 1.0 + ((Math.sqrt(5.0) + -1.0) / (2.0 / Math.cos(x)));
	double tmp;
	if (x <= -3.5e-7) {
		tmp = t_1 / (3.0 * ((Math.cos(y) * (t_2 / 2.0)) + t_3));
	} else if (x <= 0.00029) {
		tmp = ((2.0 + ((Math.sqrt(2.0) * t_0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) + (Math.sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (Math.sqrt(5.0) + (Math.cos(y) * t_2)))));
	} else {
		tmp = t_1 / (3.0 * (t_3 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = 2.0 + (t_0 * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))
	t_2 = 3.0 - math.sqrt(5.0)
	t_3 = 1.0 + ((math.sqrt(5.0) + -1.0) / (2.0 / math.cos(x)))
	tmp = 0
	if x <= -3.5e-7:
		tmp = t_1 / (3.0 * ((math.cos(y) * (t_2 / 2.0)) + t_3))
	elif x <= 0.00029:
		tmp = ((2.0 + ((math.sqrt(2.0) * t_0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) + (math.sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (math.sqrt(5.0) + (math.cos(y) * t_2)))))
	else:
		tmp = t_1 / (3.0 * (t_3 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(2.0 + Float64(t_0 * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))
	t_2 = Float64(3.0 - sqrt(5.0))
	t_3 = Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) / Float64(2.0 / cos(x))))
	tmp = 0.0
	if (x <= -3.5e-7)
		tmp = Float64(t_1 / Float64(3.0 * Float64(Float64(cos(y) * Float64(t_2 / 2.0)) + t_3)));
	elseif (x <= 0.00029)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * t_0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) + Float64(sin(x) * -0.0625))))) / 3.0) / Float64(1.0 + Float64(-0.5 + Float64(0.5 * Float64(sqrt(5.0) + Float64(cos(y) * t_2))))));
	else
		tmp = Float64(t_1 / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = 2.0 + (t_0 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))));
	t_2 = 3.0 - sqrt(5.0);
	t_3 = 1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x)));
	tmp = 0.0;
	if (x <= -3.5e-7)
		tmp = t_1 / (3.0 * ((cos(y) * (t_2 / 2.0)) + t_3));
	elseif (x <= 0.00029)
		tmp = ((2.0 + ((sqrt(2.0) * t_0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (sqrt(5.0) + (cos(y) * t_2)))));
	else
		tmp = t_1 / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(t$95$0 * 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]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(2.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e-7], N[(t$95$1 / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00029], N[(N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / N[(1.0 + N[(-0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999984e-7

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.9%

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

      1. associate-*l/ 61.9%

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

      2. sub-neg 61.9%

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

      3. metadata-eval 61.9%

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

   4. Applied egg-rr 61.9%

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

      1. metadata-eval 61.9%

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

      2. sub-neg 61.9%

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

      3. associate-/l* 61.9%

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

      4. sub-neg 61.9%

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

      5. metadata-eval 61.9%

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

   6. Simplified 61.9%

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

   if -3.49999999999999984e-7 < x < 2.9e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-lft-out 99.4%

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

      3. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if 2.9e-4 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 61.2%

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

      1. associate-*l/ 61.2%

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

      2. sub-neg 61.2%

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

      3. metadata-eval 61.2%

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

   4. Applied egg-rr 61.2%

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

      1. metadata-eval 61.2%

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

      2. sub-neg 61.2%

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

      3. associate-/l* 61.2%

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

      4. sub-neg 61.2%

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

      5. metadata-eval 61.2%

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

   6. Simplified 61.2%

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

      1. flip-- 25.9%

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

      2. metadata-eval 25.9%

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

      3. add-sqr-sqrt 25.9%

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

      4. metadata-eval 25.9%

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

   8. Applied egg-rr 61.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00029:\\ \;\;\;\;\frac{\frac{2 + \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3}}{1 + \left(-0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 10: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := \sqrt{5} + -1\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 3 - \sqrt{5}\\ t_5 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_2 \cdot t_3\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\cos x \cdot t_1 + \cos y \cdot t_4\right)\right)}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_4}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_0 \cdot t_3\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_5 - 0.5\right) + \cos y \cdot \left(1.5 - t_5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2 (* (sqrt 2.0) (sin x)))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (- 3.0 (sqrt 5.0)))
        (t_5 (/ (sqrt 5.0) 2.0)))
   (if (<= x -0.0032)
     (/
      (+ 2.0 (* t_0 (* t_2 t_3)))
      (+ 3.0 (* 3.0 (* 0.5 (+ (* (cos x) t_1) (* (cos y) t_4))))))
     (if (<= x 0.0032)
       (/
        (+
         2.0
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (sqrt 2.0) (* (- 1.0 (cos y)) (+ x (* -0.0625 (sin y)))))))
        (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) (* (cos y) (/ t_4 2.0)))))
       (/
        (+ 2.0 (* t_2 (* t_0 t_3)))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_5 0.5)) (* (cos y) (- 1.5 t_5))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = sqrt(2.0) * sin(x);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = 3.0 - sqrt(5.0);
	double t_5 = sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.0032) {
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 + (3.0 * (0.5 * ((cos(x) * t_1) + (cos(y) * t_4)))));
	} else if (x <= 0.0032) {
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_4 / 2.0))));
	} else {
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((cos(x) * (t_5 - 0.5)) + (cos(y) * (1.5 - t_5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = cos(x) - cos(y)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    t_2 = sqrt(2.0d0) * sin(x)
    t_3 = sin(y) - (sin(x) / 16.0d0)
    t_4 = 3.0d0 - sqrt(5.0d0)
    t_5 = sqrt(5.0d0) / 2.0d0
    if (x <= (-0.0032d0)) then
        tmp = (2.0d0 + (t_0 * (t_2 * t_3))) / (3.0d0 + (3.0d0 * (0.5d0 * ((cos(x) * t_1) + (cos(y) * t_4)))))
    else if (x <= 0.0032d0) then
        tmp = (2.0d0 + ((sin(y) + (sin(x) * (-0.0625d0))) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (x + ((-0.0625d0) * sin(y))))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 / 2.0d0))) + (cos(y) * (t_4 / 2.0d0))))
    else
        tmp = (2.0d0 + (t_2 * (t_0 * t_3))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_5 - 0.5d0)) + (cos(y) * (1.5d0 - t_5)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) - Math.cos(y);
	double t_1 = Math.sqrt(5.0) + -1.0;
	double t_2 = Math.sqrt(2.0) * Math.sin(x);
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_4 = 3.0 - Math.sqrt(5.0);
	double t_5 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -0.0032) {
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 + (3.0 * (0.5 * ((Math.cos(x) * t_1) + (Math.cos(y) * t_4)))));
	} else if (x <= 0.0032) {
		tmp = (2.0 + ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * (x + (-0.0625 * Math.sin(y))))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 / 2.0))) + (Math.cos(y) * (t_4 / 2.0))));
	} else {
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((Math.cos(x) * (t_5 - 0.5)) + (Math.cos(y) * (1.5 - t_5)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = math.sqrt(5.0) + -1.0
	t_2 = math.sqrt(2.0) * math.sin(x)
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	t_4 = 3.0 - math.sqrt(5.0)
	t_5 = math.sqrt(5.0) / 2.0
	tmp = 0
	if x <= -0.0032:
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 + (3.0 * (0.5 * ((math.cos(x) * t_1) + (math.cos(y) * t_4)))))
	elif x <= 0.0032:
		tmp = (2.0 + ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * (x + (-0.0625 * math.sin(y))))))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 / 2.0))) + (math.cos(y) * (t_4 / 2.0))))
	else:
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((math.cos(x) * (t_5 - 0.5)) + (math.cos(y) * (1.5 - t_5)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(sqrt(2.0) * sin(x))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(3.0 - sqrt(5.0))
	t_5 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (x <= -0.0032)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(t_2 * t_3))) / Float64(3.0 + Float64(3.0 * Float64(0.5 * Float64(Float64(cos(x) * t_1) + Float64(cos(y) * t_4))))));
	elseif (x <= 0.0032)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(x + Float64(-0.0625 * sin(y))))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(t_4 / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * t_3))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_5 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = sqrt(5.0) + -1.0;
	t_2 = sqrt(2.0) * sin(x);
	t_3 = sin(y) - (sin(x) / 16.0);
	t_4 = 3.0 - sqrt(5.0);
	t_5 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if (x <= -0.0032)
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 + (3.0 * (0.5 * ((cos(x) * t_1) + (cos(y) * t_4)))));
	elseif (x <= 0.0032)
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_4 / 2.0))));
	else
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((cos(x) * (t_5 - 0.5)) + (cos(y) * (1.5 - t_5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -0.0032], N[(N[(2.0 + N[(t$95$0 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0032], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$4 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$5 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00320000000000000015

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.3%

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

   3. Taylor expanded in x around inf 61.3%

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

      1. distribute-lft-in 61.3%

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

      2. metadata-eval 61.3%

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

      3. distribute-lft-out 61.3%

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

      4. *-commutative 61.3%

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

      5. sub-neg 61.3%

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

      6. metadata-eval 61.3%

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

      7. *-commutative 61.3%

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

   5. Simplified 61.3%

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

   if -0.00320000000000000015 < x < 0.00320000000000000015

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-lft-out 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*l* 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-lft-out 99.3%

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

   7. Simplified 99.3%

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

   if 0.00320000000000000015 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 61.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0032:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.0032:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 11: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sqrt{5} + -1\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := \cos y \cdot \frac{3 - \sqrt{5}}{2}\\ t_4 := \cos x - \cos y\\ t_5 := \sqrt{2} \cdot \sin x\\ \mathbf{if}\;x \leq -0.0062:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(t_5 \cdot t_2\right)}{3 \cdot \left(t_3 + \left(1 + \frac{t_1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0064:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_5 \cdot \left(t_4 \cdot t_2\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0))
        (t_1 (+ (sqrt 5.0) -1.0))
        (t_2 (- (sin y) (/ (sin x) 16.0)))
        (t_3 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))
        (t_4 (- (cos x) (cos y)))
        (t_5 (* (sqrt 2.0) (sin x))))
   (if (<= x -0.0062)
     (/
      (+ 2.0 (* t_4 (* t_5 t_2)))
      (* 3.0 (+ t_3 (+ 1.0 (/ t_1 (/ 2.0 (cos x)))))))
     (if (<= x 0.0064)
       (/
        (+
         2.0
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (sqrt 2.0) (* (- 1.0 (cos y)) (+ x (* -0.0625 (sin y)))))))
        (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) t_3)))
       (/
        (+ 2.0 (* t_5 (* t_4 t_2)))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = sqrt(5.0) + -1.0;
	double t_2 = sin(y) - (sin(x) / 16.0);
	double t_3 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	double t_4 = cos(x) - cos(y);
	double t_5 = sqrt(2.0) * sin(x);
	double tmp;
	if (x <= -0.0062) {
		tmp = (2.0 + (t_4 * (t_5 * t_2))) / (3.0 * (t_3 + (1.0 + (t_1 / (2.0 / cos(x))))));
	} else if (x <= 0.0064) {
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + t_3));
	} else {
		tmp = (2.0 + (t_5 * (t_4 * t_2))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = sqrt(5.0d0) + (-1.0d0)
    t_2 = sin(y) - (sin(x) / 16.0d0)
    t_3 = cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)
    t_4 = cos(x) - cos(y)
    t_5 = sqrt(2.0d0) * sin(x)
    if (x <= (-0.0062d0)) then
        tmp = (2.0d0 + (t_4 * (t_5 * t_2))) / (3.0d0 * (t_3 + (1.0d0 + (t_1 / (2.0d0 / cos(x))))))
    else if (x <= 0.0064d0) then
        tmp = (2.0d0 + ((sin(y) + (sin(x) * (-0.0625d0))) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (x + ((-0.0625d0) * sin(y))))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 / 2.0d0))) + t_3))
    else
        tmp = (2.0d0 + (t_5 * (t_4 * t_2))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = Math.sqrt(5.0) + -1.0;
	double t_2 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_3 = Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0);
	double t_4 = Math.cos(x) - Math.cos(y);
	double t_5 = Math.sqrt(2.0) * Math.sin(x);
	double tmp;
	if (x <= -0.0062) {
		tmp = (2.0 + (t_4 * (t_5 * t_2))) / (3.0 * (t_3 + (1.0 + (t_1 / (2.0 / Math.cos(x))))));
	} else if (x <= 0.0064) {
		tmp = (2.0 + ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * (x + (-0.0625 * Math.sin(y))))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 / 2.0))) + t_3));
	} else {
		tmp = (2.0 + (t_5 * (t_4 * t_2))) / (3.0 * (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = math.sqrt(5.0) + -1.0
	t_2 = math.sin(y) - (math.sin(x) / 16.0)
	t_3 = math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)
	t_4 = math.cos(x) - math.cos(y)
	t_5 = math.sqrt(2.0) * math.sin(x)
	tmp = 0
	if x <= -0.0062:
		tmp = (2.0 + (t_4 * (t_5 * t_2))) / (3.0 * (t_3 + (1.0 + (t_1 / (2.0 / math.cos(x))))))
	elif x <= 0.0064:
		tmp = (2.0 + ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * (x + (-0.0625 * math.sin(y))))))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 / 2.0))) + t_3))
	else:
		tmp = (2.0 + (t_5 * (t_4 * t_2))) / (3.0 * (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sqrt(5.0) + -1.0)
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_3 = Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))
	t_4 = Float64(cos(x) - cos(y))
	t_5 = Float64(sqrt(2.0) * sin(x))
	tmp = 0.0
	if (x <= -0.0062)
		tmp = Float64(Float64(2.0 + Float64(t_4 * Float64(t_5 * t_2))) / Float64(3.0 * Float64(t_3 + Float64(1.0 + Float64(t_1 / Float64(2.0 / cos(x)))))));
	elseif (x <= 0.0064)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(x + Float64(-0.0625 * sin(y))))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + t_3)));
	else
		tmp = Float64(Float64(2.0 + Float64(t_5 * Float64(t_4 * t_2))) / Float64(3.0 * Float64(1.0 + 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 = sqrt(5.0) + -1.0;
	t_2 = sin(y) - (sin(x) / 16.0);
	t_3 = cos(y) * ((3.0 - sqrt(5.0)) / 2.0);
	t_4 = cos(x) - cos(y);
	t_5 = sqrt(2.0) * sin(x);
	tmp = 0.0;
	if (x <= -0.0062)
		tmp = (2.0 + (t_4 * (t_5 * t_2))) / (3.0 * (t_3 + (1.0 + (t_1 / (2.0 / cos(x))))));
	elseif (x <= 0.0064)
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + t_3));
	else
		tmp = (2.0 + (t_5 * (t_4 * t_2))) / (3.0 * (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0062], N[(N[(2.0 + N[(t$95$4 * N[(t$95$5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(1.0 + N[(t$95$1 / N[(2.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0064], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$5 * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00619999999999999978

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.3%

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

      1. associate-*l/ 61.3%

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

      2. sub-neg 61.3%

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

      3. metadata-eval 61.3%

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

   4. Applied egg-rr 61.3%

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

      1. metadata-eval 61.3%

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

      2. sub-neg 61.3%

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

      3. associate-/l* 61.3%

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

      4. sub-neg 61.3%

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

      5. metadata-eval 61.3%

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

   6. Simplified 61.3%

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

   if -0.00619999999999999978 < x < 0.00640000000000000031

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-lft-out 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*l* 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-lft-out 99.3%

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

   7. Simplified 99.3%

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

   if 0.00640000000000000031 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 61.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0062:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.0064:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 12: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x - \cos y\\ t_1 := 3 - \sqrt{5}\\ t_2 := \sqrt{2} \cdot \sin x\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + t_0 \cdot \left(t_2 \cdot t_3\right)}{3 \cdot \left(\cos y \cdot \frac{t_1}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00035:\\ \;\;\;\;\frac{\frac{2 + \left(\sqrt{2} \cdot t_0\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3}}{1 + \left(-0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_2 \cdot \left(t_0 \cdot t_3\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(t_4 - 0.5\right) + \cos y \cdot \left(1.5 - t_4\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (cos x) (cos y)))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (* (sqrt 2.0) (sin x)))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (/ (sqrt 5.0) 2.0)))
   (if (<= x -3.5e-7)
     (/
      (+ 2.0 (* t_0 (* t_2 t_3)))
      (*
       3.0
       (+
        (* (cos y) (/ t_1 2.0))
        (+ 1.0 (/ (+ (sqrt 5.0) -1.0) (/ 2.0 (cos x)))))))
     (if (<= x 0.00035)
       (/
        (/
         (+
          2.0
          (*
           (* (sqrt 2.0) t_0)
           (*
            (+ (sin x) (* -0.0625 (sin y)))
            (+ (sin y) (* (sin x) -0.0625)))))
         3.0)
        (+ 1.0 (+ -0.5 (* 0.5 (+ (sqrt 5.0) (* (cos y) t_1))))))
       (/
        (+ 2.0 (* t_2 (* t_0 t_3)))
        (*
         3.0
         (+ 1.0 (+ (* (cos x) (- t_4 0.5)) (* (cos y) (- 1.5 t_4))))))))))

C

double code(double x, double y) {
	double t_0 = cos(x) - cos(y);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = sqrt(2.0) * sin(x);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -3.5e-7) {
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 * ((cos(y) * (t_1 / 2.0)) + (1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x))))));
	} else if (x <= 0.00035) {
		tmp = ((2.0 + ((sqrt(2.0) * t_0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (sqrt(5.0) + (cos(y) * t_1)))));
	} else {
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((cos(x) * (t_4 - 0.5)) + (cos(y) * (1.5 - t_4)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = cos(x) - cos(y)
    t_1 = 3.0d0 - sqrt(5.0d0)
    t_2 = sqrt(2.0d0) * sin(x)
    t_3 = sin(y) - (sin(x) / 16.0d0)
    t_4 = sqrt(5.0d0) / 2.0d0
    if (x <= (-3.5d-7)) then
        tmp = (2.0d0 + (t_0 * (t_2 * t_3))) / (3.0d0 * ((cos(y) * (t_1 / 2.0d0)) + (1.0d0 + ((sqrt(5.0d0) + (-1.0d0)) / (2.0d0 / cos(x))))))
    else if (x <= 0.00035d0) then
        tmp = ((2.0d0 + ((sqrt(2.0d0) * t_0) * ((sin(x) + ((-0.0625d0) * sin(y))) * (sin(y) + (sin(x) * (-0.0625d0)))))) / 3.0d0) / (1.0d0 + ((-0.5d0) + (0.5d0 * (sqrt(5.0d0) + (cos(y) * t_1)))))
    else
        tmp = (2.0d0 + (t_2 * (t_0 * t_3))) / (3.0d0 * (1.0d0 + ((cos(x) * (t_4 - 0.5d0)) + (cos(y) * (1.5d0 - t_4)))))
    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 = 3.0 - Math.sqrt(5.0);
	double t_2 = Math.sqrt(2.0) * Math.sin(x);
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_4 = Math.sqrt(5.0) / 2.0;
	double tmp;
	if (x <= -3.5e-7) {
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 * ((Math.cos(y) * (t_1 / 2.0)) + (1.0 + ((Math.sqrt(5.0) + -1.0) / (2.0 / Math.cos(x))))));
	} else if (x <= 0.00035) {
		tmp = ((2.0 + ((Math.sqrt(2.0) * t_0) * ((Math.sin(x) + (-0.0625 * Math.sin(y))) * (Math.sin(y) + (Math.sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (Math.sqrt(5.0) + (Math.cos(y) * t_1)))));
	} else {
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((Math.cos(x) * (t_4 - 0.5)) + (Math.cos(y) * (1.5 - t_4)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) - math.cos(y)
	t_1 = 3.0 - math.sqrt(5.0)
	t_2 = math.sqrt(2.0) * math.sin(x)
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	t_4 = math.sqrt(5.0) / 2.0
	tmp = 0
	if x <= -3.5e-7:
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 * ((math.cos(y) * (t_1 / 2.0)) + (1.0 + ((math.sqrt(5.0) + -1.0) / (2.0 / math.cos(x))))))
	elif x <= 0.00035:
		tmp = ((2.0 + ((math.sqrt(2.0) * t_0) * ((math.sin(x) + (-0.0625 * math.sin(y))) * (math.sin(y) + (math.sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (math.sqrt(5.0) + (math.cos(y) * t_1)))))
	else:
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((math.cos(x) * (t_4 - 0.5)) + (math.cos(y) * (1.5 - t_4)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) - cos(y))
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(sqrt(2.0) * sin(x))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if (x <= -3.5e-7)
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(t_2 * t_3))) / Float64(3.0 * Float64(Float64(cos(y) * Float64(t_1 / 2.0)) + Float64(1.0 + Float64(Float64(sqrt(5.0) + -1.0) / Float64(2.0 / cos(x)))))));
	elseif (x <= 0.00035)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * t_0) * Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(sin(y) + Float64(sin(x) * -0.0625))))) / 3.0) / Float64(1.0 + Float64(-0.5 + Float64(0.5 * Float64(sqrt(5.0) + Float64(cos(y) * t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_0 * t_3))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(x) * Float64(t_4 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_4))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) - cos(y);
	t_1 = 3.0 - sqrt(5.0);
	t_2 = sqrt(2.0) * sin(x);
	t_3 = sin(y) - (sin(x) / 16.0);
	t_4 = sqrt(5.0) / 2.0;
	tmp = 0.0;
	if (x <= -3.5e-7)
		tmp = (2.0 + (t_0 * (t_2 * t_3))) / (3.0 * ((cos(y) * (t_1 / 2.0)) + (1.0 + ((sqrt(5.0) + -1.0) / (2.0 / cos(x))))));
	elseif (x <= 0.00035)
		tmp = ((2.0 + ((sqrt(2.0) * t_0) * ((sin(x) + (-0.0625 * sin(y))) * (sin(y) + (sin(x) * -0.0625))))) / 3.0) / (1.0 + (-0.5 + (0.5 * (sqrt(5.0) + (cos(y) * t_1)))));
	else
		tmp = (2.0 + (t_2 * (t_0 * t_3))) / (3.0 * (1.0 + ((cos(x) * (t_4 - 0.5)) + (cos(y) * (1.5 - t_4)))));
	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[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -3.5e-7], N[(N[(2.0 + N[(t$95$0 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[(2.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00035], N[(N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / N[(1.0 + N[(-0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(t$95$2 * N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$4 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999984e-7

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 61.9%

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

      1. associate-*l/ 61.9%

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

      2. sub-neg 61.9%

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

      3. metadata-eval 61.9%

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

   4. Applied egg-rr 61.9%

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

      1. metadata-eval 61.9%

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

      2. sub-neg 61.9%

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

      3. associate-/l* 61.9%

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

      4. sub-neg 61.9%

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

      5. metadata-eval 61.9%

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

   6. Simplified 61.9%

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

   if -3.49999999999999984e-7 < x < 3.49999999999999996e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. sub-neg 99.4%

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

      2. distribute-lft-out 99.4%

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

      3. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if 3.49999999999999996e-4 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. associate-+l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

      6. *-commutative 98.9%

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

      7. div-sub 98.9%

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

      8. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in y around 0 61.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(\cos y \cdot \frac{3 - \sqrt{5}}{2} + \left(1 + \frac{\sqrt{5} + -1}{\frac{2}{\cos x}}\right)\right)}\\ \mathbf{elif}\;x \leq 0.00035:\\ \;\;\;\;\frac{\frac{2 + \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\sin y + \sin x \cdot -0.0625\right)\right)}{3}}{1 + \left(-0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot \sin x\right) \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)\right)}\\ \end{array} \]

Alternative 13: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0042 \lor \neg \left(x \leq 0.0036\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.0042) (not (<= x 0.0036)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      (* 3.0 (+ 1.0 (* 0.5 (+ (* (cos x) t_0) (* (cos y) t_1))))))
     (/
      (+
       2.0
       (*
        (+ (sin y) (* (sin x) -0.0625))
        (* (sqrt 2.0) (* (- 1.0 (cos y)) (+ x (* -0.0625 (sin y)))))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_1 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.0042) || !(x <= 0.0036)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * t_1)))));
	} else {
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 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) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = 3.0d0 - sqrt(5.0d0)
    if ((x <= (-0.0042d0)) .or. (.not. (x <= 0.0036d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 * (1.0d0 + (0.5d0 * ((cos(x) * t_0) + (cos(y) * t_1)))))
    else
        tmp = (2.0d0 + ((sin(y) + (sin(x) * (-0.0625d0))) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (x + ((-0.0625d0) * sin(y))))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * (t_1 / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if ((x <= -0.0042) || !(x <= 0.0036)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((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_0) + (Math.cos(y) * t_1)))));
	} else {
		tmp = (2.0 + ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * (x + (-0.0625 * Math.sin(y))))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * (t_1 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if (x <= -0.0042) or not (x <= 0.0036):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((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_0) + (math.cos(y) * t_1)))))
	else:
		tmp = (2.0 + ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * (x + (-0.0625 * math.sin(y))))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * (t_1 / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.0042) || !(x <= 0.0036))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 * Float64(1.0 + Float64(0.5 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(x + Float64(-0.0625 * sin(y))))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if ((x <= -0.0042) || ~((x <= 0.0036)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 * (1.0 + (0.5 * ((cos(x) * t_0) + (cos(y) * t_1)))));
	else
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0042], N[Not[LessEqual[x, 0.0036]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00419999999999999974 or 0.0035999999999999999 < 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. Taylor expanded in y around 0 61.2%

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

   3. Taylor expanded in x around inf 61.2%

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

      1. distribute-lft-out 61.2%

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

      2. *-commutative 61.2%

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

      3. sub-neg 61.2%

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

      4. metadata-eval 61.2%

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

      5. *-commutative 61.2%

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

   5. Simplified 61.2%

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

   if -0.00419999999999999974 < x < 0.0035999999999999999

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-lft-out 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*l* 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-lft-out 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0042 \lor \neg \left(x \leq 0.0036\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 14: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -0.0058 \lor \neg \left(x \leq 0.0025\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\cos x \cdot t_0 + \cos y \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_1}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -0.0058) (not (<= x 0.0025)))
     (/
      (+
       2.0
       (*
        (- (cos x) (cos y))
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
      (+ 3.0 (* 3.0 (* 0.5 (+ (* (cos x) t_0) (* (cos y) t_1))))))
     (/
      (+
       2.0
       (*
        (+ (sin y) (* (sin x) -0.0625))
        (* (sqrt 2.0) (* (- 1.0 (cos y)) (+ x (* -0.0625 (sin y)))))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_1 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -0.0058) || !(x <= 0.0025)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((cos(x) * t_0) + (cos(y) * t_1)))));
	} else {
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 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) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = 3.0d0 - sqrt(5.0d0)
    if ((x <= (-0.0058d0)) .or. (.not. (x <= 0.0025d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 + (3.0d0 * (0.5d0 * ((cos(x) * t_0) + (cos(y) * t_1)))))
    else
        tmp = (2.0d0 + ((sin(y) + (sin(x) * (-0.0625d0))) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (x + ((-0.0625d0) * sin(y))))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * (t_1 / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if ((x <= -0.0058) || !(x <= 0.0025)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((Math.cos(x) * t_0) + (Math.cos(y) * t_1)))));
	} else {
		tmp = (2.0 + ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * (x + (-0.0625 * Math.sin(y))))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * (t_1 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if (x <= -0.0058) or not (x <= 0.0025):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((math.cos(x) * t_0) + (math.cos(y) * t_1)))))
	else:
		tmp = (2.0 + ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * (x + (-0.0625 * math.sin(y))))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * (t_1 / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -0.0058) || !(x <= 0.0025))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 + Float64(3.0 * Float64(0.5 * Float64(Float64(cos(x) * t_0) + Float64(cos(y) * t_1))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * Float64(x + Float64(-0.0625 * sin(y))))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_1 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if ((x <= -0.0058) || ~((x <= 0.0025)))
		tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((cos(x) * t_0) + (cos(y) * t_1)))));
	else
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((1.0 - cos(y)) * (x + (-0.0625 * sin(y))))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_1 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0058], N[Not[LessEqual[x, 0.0025]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(3.0 * N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(x + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0058 or 0.00250000000000000005 < 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. Taylor expanded in y around 0 61.2%

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

   3. Taylor expanded in x around inf 61.2%

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

      1. distribute-lft-in 61.3%

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

      2. metadata-eval 61.3%

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

      3. distribute-lft-out 61.3%

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

      4. *-commutative 61.3%

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

      5. sub-neg 61.3%

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

      6. metadata-eval 61.3%

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

      7. *-commutative 61.3%

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

   5. Simplified 61.3%

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

   if -0.0058 < x < 0.00250000000000000005

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-lft-out 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*l* 99.3%

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

      7. *-commutative 99.3%

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

      8. distribute-lft-out 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0058 \lor \neg \left(x \leq 0.0025\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot \left(x + -0.0625 \cdot \sin y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 15: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \left(1 - \cos y\right) \cdot {\sin y}^{2}\\ \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;\frac{2 + \log \left(e^{-0.0625 \cdot \left(\sqrt{2} \cdot t_2\right)}\right)}{t_1}\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x + y \cdot -0.0625\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot t_2}{3 \cdot \left(t_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_1 (* 3.0 (+ t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
        (t_2 (* (- 1.0 (cos y)) (pow (sin y) 2.0))))
   (if (<= y -0.0054)
     (/ (+ 2.0 (log (exp (* -0.0625 (* (sqrt 2.0) t_2))))) t_1)
     (if (<= y 1.05)
       (/
        (+
         2.0
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (sqrt 2.0) (* (+ (cos x) -1.0) (+ (sin x) (* y -0.0625))))))
        t_1)
       (/
        (+ 2.0 (* (* (sqrt 2.0) -0.0625) t_2))
        (* 3.0 (+ t_0 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_1 = 3.0 * (t_0 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double t_2 = (1.0 - cos(y)) * pow(sin(y), 2.0);
	double tmp;
	if (y <= -0.0054) {
		tmp = (2.0 + log(exp((-0.0625 * (sqrt(2.0) * t_2))))) / t_1;
	} else if (y <= 1.05) {
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) + (y * -0.0625)))))) / t_1;
	} else {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * t_2)) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_1 = 3.0d0 * (t_0 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    t_2 = (1.0d0 - cos(y)) * (sin(y) ** 2.0d0)
    if (y <= (-0.0054d0)) then
        tmp = (2.0d0 + log(exp(((-0.0625d0) * (sqrt(2.0d0) * t_2))))) / t_1
    else if (y <= 1.05d0) then
        tmp = (2.0d0 + ((sin(y) + (sin(x) * (-0.0625d0))) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) + (y * (-0.0625d0))))))) / t_1
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * t_2)) / (3.0d0 * (t_0 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_1 = 3.0 * (t_0 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double t_2 = (1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0);
	double tmp;
	if (y <= -0.0054) {
		tmp = (2.0 + Math.log(Math.exp((-0.0625 * (Math.sqrt(2.0) * t_2))))) / t_1;
	} else if (y <= 1.05) {
		tmp = (2.0 + ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * (Math.sin(x) + (y * -0.0625)))))) / t_1;
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * -0.0625) * t_2)) / (3.0 * (t_0 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_1 = 3.0 * (t_0 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	t_2 = (1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)
	tmp = 0
	if y <= -0.0054:
		tmp = (2.0 + math.log(math.exp((-0.0625 * (math.sqrt(2.0) * t_2))))) / t_1
	elif y <= 1.05:
		tmp = (2.0 + ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * (math.sin(x) + (y * -0.0625)))))) / t_1
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * -0.0625) * t_2)) / (3.0 * (t_0 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_1 = Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	t_2 = Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))
	tmp = 0.0
	if (y <= -0.0054)
		tmp = Float64(Float64(2.0 + log(exp(Float64(-0.0625 * Float64(sqrt(2.0) * t_2))))) / t_1);
	elseif (y <= 1.05)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * Float64(sin(x) + Float64(y * -0.0625)))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * t_2)) / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_1 = 3.0 * (t_0 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	t_2 = (1.0 - cos(y)) * (sin(y) ^ 2.0);
	tmp = 0.0;
	if (y <= -0.0054)
		tmp = (2.0 + log(exp((-0.0625 * (sqrt(2.0) * t_2))))) / t_1;
	elseif (y <= 1.05)
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) + (y * -0.0625)))))) / t_1;
	else
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * t_2)) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(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$1 = N[(3.0 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0054], N[(N[(2.0 + N[Log[N[Exp[N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 1.05], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(y * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0054000000000000003

   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. Taylor expanded in x around 0 59.4%

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

      1. associate-*r* 59.4%

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

   4. Simplified 59.4%

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

      1. add-log-exp 59.4%

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

      2. associate-*l* 59.4%

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

   6. Applied egg-rr 59.4%

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

   if -0.0054000000000000003 < y < 1.05000000000000004

   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. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*l* 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-*l* 98.2%

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

      4. *-commutative 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. associate-*r* 98.2%

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

      7. distribute-rgt-out 98.2%

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

      8. sub-neg 98.2%

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

      9. metadata-eval 98.2%

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

      10. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if 1.05000000000000004 < y

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 57.5%

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

      1. associate-*r* 57.5%

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

   4. Simplified 57.5%

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

      1. flip-- 57.5%

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

      2. metadata-eval 57.5%

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

      3. add-sqr-sqrt 57.6%

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

      4. metadata-eval 57.6%

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

   6. Applied egg-rr 57.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0054:\\ \;\;\;\;\frac{2 + \log \left(e^{-0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x + y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 16: 79.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := 2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\\ \mathbf{if}\;y \leq -0.003:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_1 + \cos y \cdot \frac{\log \left(e^{t_0}\right)}{2}\right)}\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x + y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2
         (+
          2.0
          (* (* (sqrt 2.0) -0.0625) (* (- 1.0 (cos y)) (pow (sin y) 2.0))))))
   (if (<= y -0.003)
     (/ t_2 (* 3.0 (+ t_1 (* (cos y) (/ (log (exp t_0)) 2.0)))))
     (if (<= y 1.05)
       (/
        (+
         2.0
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (sqrt 2.0) (* (+ (cos x) -1.0) (+ (sin x) (* y -0.0625))))))
        (* 3.0 (+ t_1 (* (cos y) (/ t_0 2.0)))))
       (/
        t_2
        (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * pow(sin(y), 2.0)));
	double tmp;
	if (y <= -0.003) {
		tmp = t_2 / (3.0 * (t_1 + (cos(y) * (log(exp(t_0)) / 2.0))));
	} else if (y <= 1.05) {
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) + (y * -0.0625)))))) / (3.0 * (t_1 + (cos(y) * (t_0 / 2.0))));
	} else {
		tmp = t_2 / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_2 = 2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)))
    if (y <= (-0.003d0)) then
        tmp = t_2 / (3.0d0 * (t_1 + (cos(y) * (log(exp(t_0)) / 2.0d0))))
    else if (y <= 1.05d0) then
        tmp = (2.0d0 + ((sin(y) + (sin(x) * (-0.0625d0))) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) + (y * (-0.0625d0))))))) / (3.0d0 * (t_1 + (cos(y) * (t_0 / 2.0d0))))
    else
        tmp = t_2 / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 2.0 + ((Math.sqrt(2.0) * -0.0625) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)));
	double tmp;
	if (y <= -0.003) {
		tmp = t_2 / (3.0 * (t_1 + (Math.cos(y) * (Math.log(Math.exp(t_0)) / 2.0))));
	} else if (y <= 1.05) {
		tmp = (2.0 + ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * (Math.sin(x) + (y * -0.0625)))))) / (3.0 * (t_1 + (Math.cos(y) * (t_0 / 2.0))));
	} else {
		tmp = t_2 / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_2 = 2.0 + ((math.sqrt(2.0) * -0.0625) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)))
	tmp = 0
	if y <= -0.003:
		tmp = t_2 / (3.0 * (t_1 + (math.cos(y) * (math.log(math.exp(t_0)) / 2.0))))
	elif y <= 1.05:
		tmp = (2.0 + ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * (math.sin(x) + (y * -0.0625)))))) / (3.0 * (t_1 + (math.cos(y) * (t_0 / 2.0))))
	else:
		tmp = t_2 / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))
	tmp = 0.0
	if (y <= -0.003)
		tmp = Float64(t_2 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(log(exp(t_0)) / 2.0)))));
	elseif (y <= 1.05)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * Float64(sin(x) + Float64(y * -0.0625)))))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(t_0 / 2.0)))));
	else
		tmp = Float64(t_2 / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_2 = 2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * (sin(y) ^ 2.0)));
	tmp = 0.0;
	if (y <= -0.003)
		tmp = t_2 / (3.0 * (t_1 + (cos(y) * (log(exp(t_0)) / 2.0))));
	elseif (y <= 1.05)
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) + (y * -0.0625)))))) / (3.0 * (t_1 + (cos(y) * (t_0 / 2.0))));
	else
		tmp = t_2 / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.003], N[(t$95$2 / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(y * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / 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]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0030000000000000001

   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. Taylor expanded in x around 0 59.4%

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

      1. associate-*r* 59.4%

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

   4. Simplified 59.4%

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

      1. add-log-exp 59.4%

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

   6. Applied egg-rr 59.4%

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

   if -0.0030000000000000001 < y < 1.05000000000000004

   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. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*l* 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-*l* 98.2%

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

      4. *-commutative 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. associate-*r* 98.2%

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

      7. distribute-rgt-out 98.2%

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

      8. sub-neg 98.2%

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

      9. metadata-eval 98.2%

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

      10. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if 1.05000000000000004 < y

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 57.5%

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

      1. associate-*r* 57.5%

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

   4. Simplified 57.5%

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

      1. flip-- 57.5%

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

      2. metadata-eval 57.5%

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

      3. add-sqr-sqrt 57.6%

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

      4. metadata-eval 57.6%

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

   6. Applied egg-rr 57.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.003:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\log \left(e^{3 - \sqrt{5}}\right)}{2}\right)}\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x + y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 17: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_2 := 3 \cdot \left(t_1 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;y \leq -0.0235:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t_0\right)\right)}{t_2}\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x + y \cdot -0.0625\right)\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot t_0\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_2 (* 3.0 (+ t_1 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (<= y -0.0235)
     (/ (+ 2.0 (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 t_0)))) t_2)
     (if (<= y 1.05)
       (/
        (+
         2.0
         (*
          (+ (sin y) (* (sin x) -0.0625))
          (* (sqrt 2.0) (* (+ (cos x) -1.0) (+ (sin x) (* y -0.0625))))))
        t_2)
       (/
        (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (- 1.0 (cos y)) t_0)))
        (* 3.0 (+ t_1 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if (y <= -0.0235) {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_0)))) / t_2;
	} else if (y <= 1.05) {
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) + (y * -0.0625)))))) / t_2;
	} else {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * t_0))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(y) ** 2.0d0
    t_1 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_2 = 3.0d0 * (t_1 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if (y <= (-0.0235d0)) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((-0.0625d0) * t_0)))) / t_2
    else if (y <= 1.05d0) then
        tmp = (2.0d0 + ((sin(y) + (sin(x) * (-0.0625d0))) * (sqrt(2.0d0) * ((cos(x) + (-1.0d0)) * (sin(x) + (y * (-0.0625d0))))))) / t_2
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((1.0d0 - cos(y)) * t_0))) / (3.0d0 * (t_1 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(y), 2.0);
	double t_1 = 1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0));
	double t_2 = 3.0 * (t_1 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if (y <= -0.0235) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (-0.0625 * t_0)))) / t_2;
	} else if (y <= 1.05) {
		tmp = (2.0 + ((Math.sin(y) + (Math.sin(x) * -0.0625)) * (Math.sqrt(2.0) * ((Math.cos(x) + -1.0) * (Math.sin(x) + (y * -0.0625)))))) / t_2;
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * -0.0625) * ((1.0 - Math.cos(y)) * t_0))) / (3.0 * (t_1 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(y), 2.0)
	t_1 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_2 = 3.0 * (t_1 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if y <= -0.0235:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (-0.0625 * t_0)))) / t_2
	elif y <= 1.05:
		tmp = (2.0 + ((math.sin(y) + (math.sin(x) * -0.0625)) * (math.sqrt(2.0) * ((math.cos(x) + -1.0) * (math.sin(x) + (y * -0.0625)))))) / t_2
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * -0.0625) * ((1.0 - math.cos(y)) * t_0))) / (3.0 * (t_1 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0)))
	t_2 = Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if (y <= -0.0235)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * t_0)))) / t_2);
	elseif (y <= 1.05)
		tmp = Float64(Float64(2.0 + Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(sqrt(2.0) * Float64(Float64(cos(x) + -1.0) * Float64(sin(x) + Float64(y * -0.0625)))))) / t_2);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * t_0))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) ^ 2.0;
	t_1 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	t_2 = 3.0 * (t_1 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if (y <= -0.0235)
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_0)))) / t_2;
	elseif (y <= 1.05)
		tmp = (2.0 + ((sin(y) + (sin(x) * -0.0625)) * (sqrt(2.0) * ((cos(x) + -1.0) * (sin(x) + (y * -0.0625)))))) / t_2;
	else
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * t_0))) / (3.0 * (t_1 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[y], $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[(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]}, If[LessEqual[y, -0.0235], 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$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 1.05], N[(N[(2.0 + N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] + N[(y * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $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]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0235

   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. Taylor expanded in x around -inf 99.1%

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

   3. Taylor expanded in x around 0 59.4%

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

   if -0.0235 < y < 1.05000000000000004

   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. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*l* 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-*l* 98.2%

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

      4. *-commutative 98.2%

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

      5. distribute-lft-out 98.2%

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

      6. associate-*r* 98.2%

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

      7. distribute-rgt-out 98.2%

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

      8. sub-neg 98.2%

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

      9. metadata-eval 98.2%

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

      10. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if 1.05000000000000004 < y

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 57.5%

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

      1. associate-*r* 57.5%

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

   4. Simplified 57.5%

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

      1. flip-- 57.5%

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

      2. metadata-eval 57.5%

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

      3. add-sqr-sqrt 57.6%

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

      4. metadata-eval 57.6%

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

   6. Applied egg-rr 57.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0235:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;\frac{2 + \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot \left(\sin x + y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 18: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := \frac{4}{3 + \sqrt{5}}\\ t_2 := \sqrt{5} + -1\\ t_3 := 1 + \cos x \cdot \frac{t_2}{2}\\ t_4 := \cos x - \cos y\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(t_4 \cdot \left(-0.0625 \cdot t_0\right)\right)}{3 \cdot \left(t_3 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{2 + t_4 \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\cos x \cdot t_2 + t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot t_0\right)}{3 \cdot \left(t_3 + \cos y \cdot \frac{t_1}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (/ 4.0 (+ 3.0 (sqrt 5.0))))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (+ 1.0 (* (cos x) (/ t_2 2.0))))
        (t_4 (- (cos x) (cos y))))
   (if (<= y -5.5e-5)
     (/
      (+ 2.0 (* (sqrt 2.0) (* t_4 (* -0.0625 t_0))))
      (* 3.0 (+ t_3 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= y 0.135)
       (/
        (+ 2.0 (* t_4 (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
        (+ 3.0 (* 3.0 (* 0.5 (+ (* (cos x) t_2) t_1)))))
       (/
        (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (- 1.0 (cos y)) t_0)))
        (* 3.0 (+ t_3 (* (cos y) (/ t_1 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = 4.0 / (3.0 + sqrt(5.0));
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = 1.0 + (cos(x) * (t_2 / 2.0));
	double t_4 = cos(x) - cos(y);
	double tmp;
	if (y <= -5.5e-5) {
		tmp = (2.0 + (sqrt(2.0) * (t_4 * (-0.0625 * t_0)))) / (3.0 * (t_3 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (y <= 0.135) {
		tmp = (2.0 + (t_4 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((cos(x) * t_2) + t_1))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * t_0))) / (3.0 * (t_3 + (cos(y) * (t_1 / 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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sin(y) ** 2.0d0
    t_1 = 4.0d0 / (3.0d0 + sqrt(5.0d0))
    t_2 = sqrt(5.0d0) + (-1.0d0)
    t_3 = 1.0d0 + (cos(x) * (t_2 / 2.0d0))
    t_4 = cos(x) - cos(y)
    if (y <= (-5.5d-5)) then
        tmp = (2.0d0 + (sqrt(2.0d0) * (t_4 * ((-0.0625d0) * t_0)))) / (3.0d0 * (t_3 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else if (y <= 0.135d0) then
        tmp = (2.0d0 + (t_4 * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / (3.0d0 + (3.0d0 * (0.5d0 * ((cos(x) * t_2) + t_1))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((1.0d0 - cos(y)) * t_0))) / (3.0d0 * (t_3 + (cos(y) * (t_1 / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(y), 2.0);
	double t_1 = 4.0 / (3.0 + Math.sqrt(5.0));
	double t_2 = Math.sqrt(5.0) + -1.0;
	double t_3 = 1.0 + (Math.cos(x) * (t_2 / 2.0));
	double t_4 = Math.cos(x) - Math.cos(y);
	double tmp;
	if (y <= -5.5e-5) {
		tmp = (2.0 + (Math.sqrt(2.0) * (t_4 * (-0.0625 * t_0)))) / (3.0 * (t_3 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else if (y <= 0.135) {
		tmp = (2.0 + (t_4 * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((Math.cos(x) * t_2) + t_1))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * -0.0625) * ((1.0 - Math.cos(y)) * t_0))) / (3.0 * (t_3 + (Math.cos(y) * (t_1 / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(y), 2.0)
	t_1 = 4.0 / (3.0 + math.sqrt(5.0))
	t_2 = math.sqrt(5.0) + -1.0
	t_3 = 1.0 + (math.cos(x) * (t_2 / 2.0))
	t_4 = math.cos(x) - math.cos(y)
	tmp = 0
	if y <= -5.5e-5:
		tmp = (2.0 + (math.sqrt(2.0) * (t_4 * (-0.0625 * t_0)))) / (3.0 * (t_3 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	elif y <= 0.135:
		tmp = (2.0 + (t_4 * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((math.cos(x) * t_2) + t_1))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * -0.0625) * ((1.0 - math.cos(y)) * t_0))) / (3.0 * (t_3 + (math.cos(y) * (t_1 / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(4.0 / Float64(3.0 + sqrt(5.0)))
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0)))
	t_4 = Float64(cos(x) - cos(y))
	tmp = 0.0
	if (y <= -5.5e-5)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(t_4 * Float64(-0.0625 * t_0)))) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (y <= 0.135)
		tmp = Float64(Float64(2.0 + Float64(t_4 * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / Float64(3.0 + Float64(3.0 * Float64(0.5 * Float64(Float64(cos(x) * t_2) + t_1)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * t_0))) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(t_1 / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) ^ 2.0;
	t_1 = 4.0 / (3.0 + sqrt(5.0));
	t_2 = sqrt(5.0) + -1.0;
	t_3 = 1.0 + (cos(x) * (t_2 / 2.0));
	t_4 = cos(x) - cos(y);
	tmp = 0.0;
	if (y <= -5.5e-5)
		tmp = (2.0 + (sqrt(2.0) * (t_4 * (-0.0625 * t_0)))) / (3.0 * (t_3 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	elseif (y <= 0.135)
		tmp = (2.0 + (t_4 * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / (3.0 + (3.0 * (0.5 * ((cos(x) * t_2) + t_1))));
	else
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * t_0))) / (3.0 * (t_3 + (cos(y) * (t_1 / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e-5], N[(N[(2.0 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 * N[(-0.0625 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.135], N[(N[(2.0 + N[(t$95$4 * 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[(3.0 * N[(0.5 * N[(N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5000000000000002e-5

   1. Initial program 99.1%

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

   2. Taylor expanded in x around -inf 99.1%

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

   3. Taylor expanded in x around 0 59.4%

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

   if -5.5000000000000002e-5 < y < 0.13500000000000001

   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. Taylor expanded in y around 0 98.4%

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

   3. Taylor expanded in y around 0 98.3%

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

      1. distribute-rgt-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. distribute-lft-out 98.4%

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

      4. *-commutative 98.4%

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

      5. sub-neg 98.4%

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

      6. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

      1. flip-- 67.1%

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

      2. metadata-eval 67.1%

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

      3. add-sqr-sqrt 67.1%

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

      4. metadata-eval 67.1%

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

   7. Applied egg-rr 98.5%

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

   if 0.13500000000000001 < y

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 57.0%

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

      1. associate-*r* 57.0%

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

   4. Simplified 57.0%

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

      1. flip-- 56.9%

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

      2. metadata-eval 56.9%

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

      3. add-sqr-sqrt 57.1%

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

      4. metadata-eval 57.1%

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

   6. Applied egg-rr 57.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \frac{4}{3 + \sqrt{5}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 19: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := {\sin y}^{2}\\ t_2 := \sqrt{5} + -1\\ t_3 := 1 + \cos x \cdot \frac{t_2}{2}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot t_1\right)\right)}{3 \cdot \left(t_3 + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(t_0 + \cos x \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot t_1\right)}{3 \cdot \left(t_3 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0)))
        (t_1 (pow (sin y) 2.0))
        (t_2 (+ (sqrt 5.0) -1.0))
        (t_3 (+ 1.0 (* (cos x) (/ t_2 2.0)))))
   (if (<= y -5.2e-5)
     (/
      (+ 2.0 (* (sqrt 2.0) (* (- (cos x) (cos y)) (* -0.0625 t_1))))
      (* 3.0 (+ t_3 (* (cos y) (/ t_0 2.0)))))
     (if (<= y 0.135)
       (/
        (+
         2.0
         (*
          (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
          (+ (cos x) -1.0)))
        (+ 3.0 (* 3.0 (* 0.5 (+ t_0 (* (cos x) t_2))))))
       (/
        (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (- 1.0 (cos y)) t_1)))
        (* 3.0 (+ t_3 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = pow(sin(y), 2.0);
	double t_2 = sqrt(5.0) + -1.0;
	double t_3 = 1.0 + (cos(x) * (t_2 / 2.0));
	double tmp;
	if (y <= -5.2e-5) {
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_1)))) / (3.0 * (t_3 + (cos(y) * (t_0 / 2.0))));
	} else if (y <= 0.135) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * (t_0 + (cos(x) * t_2)))));
	} else {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * t_1))) / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = sin(y) ** 2.0d0
    t_2 = sqrt(5.0d0) + (-1.0d0)
    t_3 = 1.0d0 + (cos(x) * (t_2 / 2.0d0))
    if (y <= (-5.2d-5)) then
        tmp = (2.0d0 + (sqrt(2.0d0) * ((cos(x) - cos(y)) * ((-0.0625d0) * t_1)))) / (3.0d0 * (t_3 + (cos(y) * (t_0 / 2.0d0))))
    else if (y <= 0.135d0) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) + (-1.0d0)))) / (3.0d0 + (3.0d0 * (0.5d0 * (t_0 + (cos(x) * t_2)))))
    else
        tmp = (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((1.0d0 - cos(y)) * t_1))) / (3.0d0 * (t_3 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 - Math.sqrt(5.0);
	double t_1 = Math.pow(Math.sin(y), 2.0);
	double t_2 = Math.sqrt(5.0) + -1.0;
	double t_3 = 1.0 + (Math.cos(x) * (t_2 / 2.0));
	double tmp;
	if (y <= -5.2e-5) {
		tmp = (2.0 + (Math.sqrt(2.0) * ((Math.cos(x) - Math.cos(y)) * (-0.0625 * t_1)))) / (3.0 * (t_3 + (Math.cos(y) * (t_0 / 2.0))));
	} else if (y <= 0.135) {
		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 + (3.0 * (0.5 * (t_0 + (Math.cos(x) * t_2)))));
	} else {
		tmp = (2.0 + ((Math.sqrt(2.0) * -0.0625) * ((1.0 - Math.cos(y)) * t_1))) / (3.0 * (t_3 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.pow(math.sin(y), 2.0)
	t_2 = math.sqrt(5.0) + -1.0
	t_3 = 1.0 + (math.cos(x) * (t_2 / 2.0))
	tmp = 0
	if y <= -5.2e-5:
		tmp = (2.0 + (math.sqrt(2.0) * ((math.cos(x) - math.cos(y)) * (-0.0625 * t_1)))) / (3.0 * (t_3 + (math.cos(y) * (t_0 / 2.0))))
	elif y <= 0.135:
		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 + (3.0 * (0.5 * (t_0 + (math.cos(x) * t_2)))))
	else:
		tmp = (2.0 + ((math.sqrt(2.0) * -0.0625) * ((1.0 - math.cos(y)) * t_1))) / (3.0 * (t_3 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = sin(y) ^ 2.0
	t_2 = Float64(sqrt(5.0) + -1.0)
	t_3 = Float64(1.0 + Float64(cos(x) * Float64(t_2 / 2.0)))
	tmp = 0.0
	if (y <= -5.2e-5)
		tmp = Float64(Float64(2.0 + Float64(sqrt(2.0) * Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * t_1)))) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(t_0 / 2.0)))));
	elseif (y <= 0.135)
		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(3.0 * Float64(0.5 * Float64(t_0 + Float64(cos(x) * t_2))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * t_1))) / Float64(3.0 * Float64(t_3 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = sin(y) ^ 2.0;
	t_2 = sqrt(5.0) + -1.0;
	t_3 = 1.0 + (cos(x) * (t_2 / 2.0));
	tmp = 0.0;
	if (y <= -5.2e-5)
		tmp = (2.0 + (sqrt(2.0) * ((cos(x) - cos(y)) * (-0.0625 * t_1)))) / (3.0 * (t_3 + (cos(y) * (t_0 / 2.0))));
	elseif (y <= 0.135)
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * (t_0 + (cos(x) * t_2)))));
	else
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * t_1))) / (3.0 * (t_3 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e-5], 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$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.135], 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[(3.0 * N[(0.5 * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$3 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.19999999999999968e-5

   1. Initial program 99.1%

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

   2. Taylor expanded in x around -inf 99.1%

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

   3. Taylor expanded in x around 0 59.4%

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

   if -5.19999999999999968e-5 < y < 0.13500000000000001

   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. Taylor expanded in y around 0 98.4%

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

   3. Taylor expanded in y around 0 98.3%

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

      1. distribute-rgt-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. distribute-lft-out 98.4%

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

      4. *-commutative 98.4%

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

      5. sub-neg 98.4%

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

      6. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in y around 0 98.4%

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

   if 0.13500000000000001 < y

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 57.0%

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

      1. associate-*r* 57.0%

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

   4. Simplified 57.0%

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

      1. flip-- 56.9%

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

      2. metadata-eval 56.9%

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

      3. add-sqr-sqrt 57.1%

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

      4. metadata-eval 57.1%

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

   6. Applied egg-rr 57.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]

Alternative 20: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1 \cdot 10^{-5} \lor \neg \left(y \leq 0.135\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot t_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -1e-5) (not (<= y 0.135)))
     (/
      (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (- 1.0 (cos y)) (pow (sin y) 2.0))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ t_0 2.0)))
        (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
        (+ (cos x) -1.0)))
      (+ 3.0 (* 3.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) (* (cos x) t_0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1e-5) || !(y <= 0.135)) {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * pow(sin(y), 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    if ((y <= (-1d-5)) .or. (.not. (y <= 0.135d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    else
        tmp = (2.0d0 + (((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) + (-1.0d0)))) / (3.0d0 + (3.0d0 * (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1e-5) || !(y <= 0.135)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * -0.0625) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	} else {
		tmp = (2.0 + (((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))) * (Math.cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * ((3.0 - Math.sqrt(5.0)) + (Math.cos(x) * t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (y <= -1e-5) or not (y <= 0.135):
		tmp = (2.0 + ((math.sqrt(2.0) * -0.0625) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	else:
		tmp = (2.0 + (((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))) * (math.cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * ((3.0 - math.sqrt(5.0)) + (math.cos(x) * t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -1e-5) || !(y <= 0.135))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))) * Float64(cos(x) + -1.0))) / Float64(3.0 + Float64(3.0 * Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(cos(x) * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((y <= -1e-5) || ~((y <= 0.135)))
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * (sin(y) ^ 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	else
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -1e-5], N[Not[LessEqual[y, 0.135]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(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[(3.0 * N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000008e-5 or 0.13500000000000001 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 58.2%

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

      1. associate-*r* 58.2%

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

   4. Simplified 58.2%

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

      1. flip-- 58.2%

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

      2. metadata-eval 58.2%

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

      3. add-sqr-sqrt 58.3%

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

      4. metadata-eval 58.3%

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

   6. Applied egg-rr 58.3%

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

   if -1.00000000000000008e-5 < y < 0.13500000000000001

   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. Taylor expanded in y around 0 98.4%

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

   3. Taylor expanded in y around 0 98.3%

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

      1. distribute-rgt-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. distribute-lft-out 98.4%

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

      4. *-commutative 98.4%

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

      5. sub-neg 98.4%

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

      6. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in y around 0 98.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-5} \lor \neg \left(y \leq 0.135\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \end{array} \]

Alternative 21: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;y \leq -1.22 \cdot 10^{-5} \lor \neg \left(y \leq 0.135\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_1}{2}\right) + \cos y \cdot \frac{t_0}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(t_0 + \cos x \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= y -1.22e-5) (not (<= y 0.135)))
     (/
      (+ 2.0 (* (* (sqrt 2.0) -0.0625) (* (- 1.0 (cos y)) (pow (sin y) 2.0))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_1 2.0))) (* (cos y) (/ t_0 2.0)))))
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
        (+ (cos x) -1.0)))
      (+ 3.0 (* 3.0 (* 0.5 (+ t_0 (* (cos x) t_1)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1.22e-5) || !(y <= 0.135)) {
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * pow(sin(y), 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	} else {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * (t_0 + (cos(x) * 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 = 3.0d0 - sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((y <= (-1.22d-5)) .or. (.not. (y <= 0.135d0))) then
        tmp = (2.0d0 + ((sqrt(2.0d0) * (-0.0625d0)) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0)))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_1 / 2.0d0))) + (cos(y) * (t_0 / 2.0d0))))
    else
        tmp = (2.0d0 + (((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) + (-1.0d0)))) / (3.0d0 + (3.0d0 * (0.5d0 * (t_0 + (cos(x) * 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 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((y <= -1.22e-5) || !(y <= 0.135)) {
		tmp = (2.0 + ((Math.sqrt(2.0) * -0.0625) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0)))) / (3.0 * ((1.0 + (Math.cos(x) * (t_1 / 2.0))) + (Math.cos(y) * (t_0 / 2.0))));
	} else {
		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 + (3.0 * (0.5 * (t_0 + (Math.cos(x) * t_1)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (y <= -1.22e-5) or not (y <= 0.135):
		tmp = (2.0 + ((math.sqrt(2.0) * -0.0625) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0)))) / (3.0 * ((1.0 + (math.cos(x) * (t_1 / 2.0))) + (math.cos(y) * (t_0 / 2.0))))
	else:
		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 + (3.0 * (0.5 * (t_0 + (math.cos(x) * t_1)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((y <= -1.22e-5) || !(y <= 0.135))
		tmp = Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * -0.0625) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 / 2.0))) + Float64(cos(y) * Float64(t_0 / 2.0)))));
	else
		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(3.0 * Float64(0.5 * Float64(t_0 + Float64(cos(x) * t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((y <= -1.22e-5) || ~((y <= 0.135)))
		tmp = (2.0 + ((sqrt(2.0) * -0.0625) * ((1.0 - cos(y)) * (sin(y) ^ 2.0)))) / (3.0 * ((1.0 + (cos(x) * (t_1 / 2.0))) + (cos(y) * (t_0 / 2.0))));
	else
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * (t_0 + (cos(x) * 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[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[y, -1.22e-5], N[Not[LessEqual[y, 0.135]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(3.0 * N[(0.5 * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.22000000000000001e-5 or 0.13500000000000001 < y

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 58.2%

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

      1. associate-*r* 58.2%

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

   4. Simplified 58.2%

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

   if -1.22000000000000001e-5 < y < 0.13500000000000001

   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. Taylor expanded in y around 0 98.4%

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

   3. Taylor expanded in y around 0 98.3%

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

      1. distribute-rgt-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. distribute-lft-out 98.4%

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

      4. *-commutative 98.4%

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

      5. sub-neg 98.4%

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

      6. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in y around 0 98.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{-5} \lor \neg \left(y \leq 0.135\right):\\ \;\;\;\;\frac{2 + \left(\sqrt{2} \cdot -0.0625\right) \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \end{array} \]

Alternative 22: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ t_1 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-7} \lor \neg \left(x \leq 11600\right):\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(t_0 + \cos x \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\cos y \cdot t_0\right) + 0.5 \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))) (t_1 (+ (sqrt 5.0) -1.0)))
   (if (or (<= x -3.5e-7) (not (<= x 11600.0)))
     (/
      (+
       2.0
       (*
        (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))
        (+ (cos x) -1.0)))
      (+ 3.0 (* 3.0 (* 0.5 (+ t_0 (* (cos x) t_1))))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
       (+ 1.0 (+ (* 0.5 (* (cos y) t_0)) (* 0.5 t_1))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double t_1 = sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -3.5e-7) || !(x <= 11600.0)) {
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * (t_0 + (cos(x) * t_1)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (1.0 + ((0.5 * (cos(y) * t_0)) + (0.5 * t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 - sqrt(5.0d0)
    t_1 = sqrt(5.0d0) + (-1.0d0)
    if ((x <= (-3.5d-7)) .or. (.not. (x <= 11600.0d0))) then
        tmp = (2.0d0 + (((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))) * (cos(x) + (-1.0d0)))) / (3.0d0 + (3.0d0 * (0.5d0 * (t_0 + (cos(x) * t_1)))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (1.0d0 + ((0.5d0 * (cos(y) * t_0)) + (0.5d0 * 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 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if ((x <= -3.5e-7) || !(x <= 11600.0)) {
		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 + (3.0 * (0.5 * (t_0 + (Math.cos(x) * t_1)))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (1.0 + ((0.5 * (Math.cos(y) * t_0)) + (0.5 * t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	t_1 = math.sqrt(5.0) + -1.0
	tmp = 0
	if (x <= -3.5e-7) or not (x <= 11600.0):
		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 + (3.0 * (0.5 * (t_0 + (math.cos(x) * t_1)))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (1.0 + ((0.5 * (math.cos(y) * t_0)) + (0.5 * t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	t_1 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if ((x <= -3.5e-7) || !(x <= 11600.0))
		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(3.0 * Float64(0.5 * Float64(t_0 + Float64(cos(x) * t_1))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(y) * t_0)) + Float64(0.5 * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	t_1 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if ((x <= -3.5e-7) || ~((x <= 11600.0)))
		tmp = (2.0 + (((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))) * (cos(x) + -1.0))) / (3.0 + (3.0 * (0.5 * (t_0 + (cos(x) * t_1)))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (1.0 + ((0.5 * (cos(y) * t_0)) + (0.5 * 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[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[Or[LessEqual[x, -3.5e-7], N[Not[LessEqual[x, 11600.0]], $MachinePrecision]], 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[(3.0 * N[(0.5 * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999984e-7 or 11600 < 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. Taylor expanded in y around 0 62.1%

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

   3. Taylor expanded in y around 0 58.3%

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

      1. distribute-rgt-in 58.3%

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

      2. metadata-eval 58.3%

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

      3. distribute-lft-out 58.3%

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

      4. *-commutative 58.3%

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

      5. sub-neg 58.3%

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

      6. metadata-eval 58.3%

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

   5. Simplified 58.3%

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

   6. Taylor expanded in y around 0 58.5%

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

   if -3.49999999999999984e-7 < x < 11600

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   3. Taylor expanded in x around 0 97.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-7} \lor \neg \left(x \leq 11600\right):\\ \;\;\;\;\frac{2 + \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x + -1\right)}{3 + 3 \cdot \left(0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 23: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-7} \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)\right)}{1 + 0.5 \cdot \left(t_1 + \cos x \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\cos y \cdot t_1\right) + 0.5 \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0)) (t_1 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -3.5e-7) (not (<= x 0.00031)))
     (/
      (*
       0.3333333333333333
       (+
        2.0
        (* -0.0625 (* (+ (cos x) -1.0) (* (sqrt 2.0) (pow (sin x) 2.0))))))
      (+ 1.0 (* 0.5 (+ t_1 (* (cos x) t_0)))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
       (+ 1.0 (+ (* 0.5 (* (cos y) t_1)) (* 0.5 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -3.5e-7) || !(x <= 0.00031)) {
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((cos(x) + -1.0) * (sqrt(2.0) * pow(sin(x), 2.0)))))) / (1.0 + (0.5 * (t_1 + (cos(x) * t_0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (1.0 + ((0.5 * (cos(y) * t_1)) + (0.5 * 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) :: tmp
    t_0 = sqrt(5.0d0) + (-1.0d0)
    t_1 = 3.0d0 - sqrt(5.0d0)
    if ((x <= (-3.5d-7)) .or. (.not. (x <= 0.00031d0))) then
        tmp = (0.3333333333333333d0 * (2.0d0 + ((-0.0625d0) * ((cos(x) + (-1.0d0)) * (sqrt(2.0d0) * (sin(x) ** 2.0d0)))))) / (1.0d0 + (0.5d0 * (t_1 + (cos(x) * t_0))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (1.0d0 + ((0.5d0 * (cos(y) * t_1)) + (0.5d0 * t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) + -1.0;
	double t_1 = 3.0 - Math.sqrt(5.0);
	double tmp;
	if ((x <= -3.5e-7) || !(x <= 0.00031)) {
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((Math.cos(x) + -1.0) * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0)))))) / (1.0 + (0.5 * (t_1 + (Math.cos(x) * t_0))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (1.0 + ((0.5 * (Math.cos(y) * t_1)) + (0.5 * t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) + -1.0
	t_1 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if (x <= -3.5e-7) or not (x <= 0.00031):
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((math.cos(x) + -1.0) * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0)))))) / (1.0 + (0.5 * (t_1 + (math.cos(x) * t_0))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (1.0 + ((0.5 * (math.cos(y) * t_1)) + (0.5 * t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -3.5e-7) || !(x <= 0.00031))
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * (sin(x) ^ 2.0)))))) / Float64(1.0 + Float64(0.5 * Float64(t_1 + Float64(cos(x) * t_0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(y) * t_1)) + Float64(0.5 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) + -1.0;
	t_1 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if ((x <= -3.5e-7) || ~((x <= 0.00031)))
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((cos(x) + -1.0) * (sqrt(2.0) * (sin(x) ^ 2.0)))))) / (1.0 + (0.5 * (t_1 + (cos(x) * t_0))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (1.0 + ((0.5 * (cos(y) * t_1)) + (0.5 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -3.5e-7], N[Not[LessEqual[x, 0.00031]], $MachinePrecision]], N[(N[(0.3333333333333333 * N[(2.0 + N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999984e-7 or 3.1e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around inf 98.9%

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

   3. Taylor expanded in y around 0 57.2%

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

      1. associate-*r/ 57.3%

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

      2. associate-*r* 57.3%

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

      3. sub-neg 57.3%

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

      4. metadata-eval 57.3%

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

      5. distribute-lft-out 57.3%

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

      6. *-commutative 57.3%

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

      7. sub-neg 57.3%

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

      8. metadata-eval 57.3%

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

   5. Simplified 57.3%

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

   if -3.49999999999999984e-7 < x < 3.1e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   3. Taylor expanded in x around 0 99.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-7} \lor \neg \left(x \leq 0.00031\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \end{array} \]

Alternative 24: 79.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := 3 - \sqrt{5}\\ t_2 := \cos x + -1\\ t_3 := \sqrt{5} + -1\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(t_2 \cdot \left(\sqrt{2} \cdot t_0\right)\right)\right)}{1 + 0.5 \cdot \left(t_1 + \cos x \cdot t_3\right)}\\ \mathbf{elif}\;x \leq 0.00027:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\cos y \cdot t_1\right) + 0.5 \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(t_2 \cdot t_0\right)\right)}{t_1 \cdot 1.5 + 3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (- 3.0 (sqrt 5.0)))
        (t_2 (+ (cos x) -1.0))
        (t_3 (+ (sqrt 5.0) -1.0)))
   (if (<= x -3.5e-7)
     (/
      (* 0.3333333333333333 (+ 2.0 (* -0.0625 (* t_2 (* (sqrt 2.0) t_0)))))
      (+ 1.0 (* 0.5 (+ t_1 (* (cos x) t_3)))))
     (if (<= x 0.00027)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
         (+ 1.0 (+ (* 0.5 (* (cos y) t_1)) (* 0.5 t_3)))))
       (/
        (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* t_2 t_0))))
        (+
         (* t_1 1.5)
         (* 3.0 (+ 1.0 (* (cos x) (- (* 0.5 (sqrt 5.0)) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = 3.0 - sqrt(5.0);
	double t_2 = cos(x) + -1.0;
	double t_3 = sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -3.5e-7) {
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_0))))) / (1.0 + (0.5 * (t_1 + (cos(x) * t_3))));
	} else if (x <= 0.00027) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / (1.0 + ((0.5 * (cos(y) * t_1)) + (0.5 * t_3))));
	} else {
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * (t_2 * t_0)))) / ((t_1 * 1.5) + (3.0 * (1.0 + (cos(x) * ((0.5 * sqrt(5.0)) - 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = 3.0d0 - sqrt(5.0d0)
    t_2 = cos(x) + (-1.0d0)
    t_3 = sqrt(5.0d0) + (-1.0d0)
    if (x <= (-3.5d-7)) then
        tmp = (0.3333333333333333d0 * (2.0d0 + ((-0.0625d0) * (t_2 * (sqrt(2.0d0) * t_0))))) / (1.0d0 + (0.5d0 * (t_1 + (cos(x) * t_3))))
    else if (x <= 0.00027d0) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / (1.0d0 + ((0.5d0 * (cos(y) * t_1)) + (0.5d0 * t_3))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * (t_2 * t_0)))) / ((t_1 * 1.5d0) + (3.0d0 * (1.0d0 + (cos(x) * ((0.5d0 * sqrt(5.0d0)) - 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = 3.0 - Math.sqrt(5.0);
	double t_2 = Math.cos(x) + -1.0;
	double t_3 = Math.sqrt(5.0) + -1.0;
	double tmp;
	if (x <= -3.5e-7) {
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * (t_2 * (Math.sqrt(2.0) * t_0))))) / (1.0 + (0.5 * (t_1 + (Math.cos(x) * t_3))));
	} else if (x <= 0.00027) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / (1.0 + ((0.5 * (Math.cos(y) * t_1)) + (0.5 * t_3))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.sqrt(2.0) * (t_2 * t_0)))) / ((t_1 * 1.5) + (3.0 * (1.0 + (Math.cos(x) * ((0.5 * Math.sqrt(5.0)) - 0.5)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = 3.0 - math.sqrt(5.0)
	t_2 = math.cos(x) + -1.0
	t_3 = math.sqrt(5.0) + -1.0
	tmp = 0
	if x <= -3.5e-7:
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * (t_2 * (math.sqrt(2.0) * t_0))))) / (1.0 + (0.5 * (t_1 + (math.cos(x) * t_3))))
	elif x <= 0.00027:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / (1.0 + ((0.5 * (math.cos(y) * t_1)) + (0.5 * t_3))))
	else:
		tmp = (2.0 + (-0.0625 * (math.sqrt(2.0) * (t_2 * t_0)))) / ((t_1 * 1.5) + (3.0 * (1.0 + (math.cos(x) * ((0.5 * math.sqrt(5.0)) - 0.5)))))
	return tmp

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(3.0 - sqrt(5.0))
	t_2 = Float64(cos(x) + -1.0)
	t_3 = Float64(sqrt(5.0) + -1.0)
	tmp = 0.0
	if (x <= -3.5e-7)
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64(t_2 * Float64(sqrt(2.0) * t_0))))) / Float64(1.0 + Float64(0.5 * Float64(t_1 + Float64(cos(x) * t_3)))));
	elseif (x <= 0.00027)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(1.0 + Float64(Float64(0.5 * Float64(cos(y) * t_1)) + Float64(0.5 * t_3)))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(t_2 * t_0)))) / Float64(Float64(t_1 * 1.5) + Float64(3.0 * Float64(1.0 + Float64(cos(x) * Float64(Float64(0.5 * sqrt(5.0)) - 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) ^ 2.0;
	t_1 = 3.0 - sqrt(5.0);
	t_2 = cos(x) + -1.0;
	t_3 = sqrt(5.0) + -1.0;
	tmp = 0.0;
	if (x <= -3.5e-7)
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * (t_2 * (sqrt(2.0) * t_0))))) / (1.0 + (0.5 * (t_1 + (cos(x) * t_3))));
	elseif (x <= 0.00027)
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / (1.0 + ((0.5 * (cos(y) * t_1)) + (0.5 * t_3))));
	else
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * (t_2 * t_0)))) / ((t_1 * 1.5) + (3.0 * (1.0 + (cos(x) * ((0.5 * sqrt(5.0)) - 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -3.5e-7], N[(N[(0.3333333333333333 * N[(2.0 + N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$1 + N[(N[Cos[x], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00027], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.5 * N[(N[Cos[y], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * 1.5), $MachinePrecision] + N[(3.0 * N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999984e-7

   1. Initial program 98.8%

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

   2. Taylor expanded in y around inf 98.9%

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

   3. Taylor expanded in y around 0 57.0%

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

      1. associate-*r/ 57.1%

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

      2. associate-*r* 57.1%

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

      3. sub-neg 57.1%

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

      4. metadata-eval 57.1%

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

      5. distribute-lft-out 57.1%

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

      6. *-commutative 57.1%

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

      7. sub-neg 57.1%

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

      8. metadata-eval 57.1%

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

   5. Simplified 57.1%

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

   if -3.49999999999999984e-7 < x < 2.70000000000000003e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   3. Taylor expanded in x around 0 99.1%

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

   if 2.70000000000000003e-4 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around inf 99.0%

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

      1. distribute-rgt-in 99.0%

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

      2. *-commutative 99.0%

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

      3. div-sub 99.0%

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

      4. metadata-eval 99.0%

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

      5. *-commutative 99.0%

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

      6. div-inv 99.0%

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

      7. metadata-eval 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in y around 0 57.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{elif}\;x \leq 0.00027:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1 + \left(0.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 0.5 \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(\cos x + -1\right) \cdot {\sin x}^{2}\right)\right)}{\left(3 - \sqrt{5}\right) \cdot 1.5 + 3 \cdot \left(1 + \cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right)\right)}\\ \end{array} \]

Alternative 25: 79.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-7} \lor \neg \left(x \leq 0.00027\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)\right)}{1 + 0.5 \cdot \left(t_0 + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\cos y \cdot t_0\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -3.1e-7) (not (<= x 0.00027)))
     (/
      (*
       0.3333333333333333
       (+
        2.0
        (* -0.0625 (* (+ (cos x) -1.0) (* (sqrt 2.0) (pow (sin x) 2.0))))))
      (+ 1.0 (* 0.5 (+ t_0 (* (cos x) (+ (sqrt 5.0) -1.0))))))
     (/
      (+ 2.0 (* -0.0625 (* (sqrt 2.0) (* (- 1.0 (cos y)) (pow (sin y) 2.0)))))
      (+ (* 1.5 (* (cos y) t_0)) (* 3.0 (+ 0.5 (* 0.5 (sqrt 5.0)))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -3.1e-7) || !(x <= 0.00027)) {
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((cos(x) + -1.0) * (sqrt(2.0) * pow(sin(x), 2.0)))))) / (1.0 + (0.5 * (t_0 + (cos(x) * (sqrt(5.0) + -1.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * pow(sin(y), 2.0))))) / ((1.5 * (cos(y) * t_0)) + (3.0 * (0.5 + (0.5 * 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 = 3.0d0 - sqrt(5.0d0)
    if ((x <= (-3.1d-7)) .or. (.not. (x <= 0.00027d0))) then
        tmp = (0.3333333333333333d0 * (2.0d0 + ((-0.0625d0) * ((cos(x) + (-1.0d0)) * (sqrt(2.0d0) * (sin(x) ** 2.0d0)))))) / (1.0d0 + (0.5d0 * (t_0 + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * (sqrt(2.0d0) * ((1.0d0 - cos(y)) * (sin(y) ** 2.0d0))))) / ((1.5d0 * (cos(y) * t_0)) + (3.0d0 * (0.5d0 + (0.5d0 * sqrt(5.0d0)))))
    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 tmp;
	if ((x <= -3.1e-7) || !(x <= 0.00027)) {
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((Math.cos(x) + -1.0) * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0)))))) / (1.0 + (0.5 * (t_0 + (Math.cos(x) * (Math.sqrt(5.0) + -1.0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.sqrt(2.0) * ((1.0 - Math.cos(y)) * Math.pow(Math.sin(y), 2.0))))) / ((1.5 * (Math.cos(y) * t_0)) + (3.0 * (0.5 + (0.5 * Math.sqrt(5.0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 - math.sqrt(5.0)
	tmp = 0
	if (x <= -3.1e-7) or not (x <= 0.00027):
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((math.cos(x) + -1.0) * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0)))))) / (1.0 + (0.5 * (t_0 + (math.cos(x) * (math.sqrt(5.0) + -1.0)))))
	else:
		tmp = (2.0 + (-0.0625 * (math.sqrt(2.0) * ((1.0 - math.cos(y)) * math.pow(math.sin(y), 2.0))))) / ((1.5 * (math.cos(y) * t_0)) + (3.0 * (0.5 + (0.5 * math.sqrt(5.0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -3.1e-7) || !(x <= 0.00027))
		tmp = Float64(Float64(0.3333333333333333 * Float64(2.0 + Float64(-0.0625 * Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * (sin(x) ^ 2.0)))))) / Float64(1.0 + Float64(0.5 * Float64(t_0 + Float64(cos(x) * Float64(sqrt(5.0) + -1.0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(sqrt(2.0) * Float64(Float64(1.0 - cos(y)) * (sin(y) ^ 2.0))))) / Float64(Float64(1.5 * Float64(cos(y) * t_0)) + Float64(3.0 * Float64(0.5 + Float64(0.5 * sqrt(5.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 - sqrt(5.0);
	tmp = 0.0;
	if ((x <= -3.1e-7) || ~((x <= 0.00027)))
		tmp = (0.3333333333333333 * (2.0 + (-0.0625 * ((cos(x) + -1.0) * (sqrt(2.0) * (sin(x) ^ 2.0)))))) / (1.0 + (0.5 * (t_0 + (cos(x) * (sqrt(5.0) + -1.0)))));
	else
		tmp = (2.0 + (-0.0625 * (sqrt(2.0) * ((1.0 - cos(y)) * (sin(y) ^ 2.0))))) / ((1.5 * (cos(y) * t_0)) + (3.0 * (0.5 + (0.5 * sqrt(5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -3.1e-7], N[Not[LessEqual[x, 0.00027]], $MachinePrecision]], N[(N[(0.3333333333333333 * N[(2.0 + N[(-0.0625 * N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 * N[(t$95$0 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(0.5 + N[(0.5 * N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e-7 or 2.70000000000000003e-4 < x

   1. Initial program 98.9%

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

   2. Taylor expanded in y around inf 98.9%

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

   3. Taylor expanded in y around 0 57.2%

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

      1. associate-*r/ 57.3%

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

      2. associate-*r* 57.3%

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

      3. sub-neg 57.3%

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

      4. metadata-eval 57.3%

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

      5. distribute-lft-out 57.3%

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

      6. *-commutative 57.3%

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

      7. sub-neg 57.3%

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

      8. metadata-eval 57.3%

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

   5. Simplified 57.3%

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

   if -3.1e-7 < x < 2.70000000000000003e-4

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

      1. distribute-rgt-in 99.6%

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

      2. *-commutative 99.6%

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

      3. div-sub 99.6%

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

      4. metadata-eval 99.6%

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

      5. *-commutative 99.6%

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

      6. div-inv 99.6%

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

      7. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 99.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-7} \lor \neg \left(x \leq 0.00027\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(2 + -0.0625 \cdot \left(\left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)\right)}{1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\sqrt{2} \cdot \left(\left(1 - \cos y\right) \cdot {\sin y}^{2}\right)\right)}{1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 3 \cdot \left(0.5 + 0.5 \cdot \sqrt{5}\right)}\\ \end{array} \]

Alternative 26: 60.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.3333333333333333d0 * (2.0d0 + ((-0.0625d0) * ((cos(x) + (-1.0d0)) * (sqrt(2.0d0) * (sin(x) ** 2.0d0)))))) / (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * (sqrt(5.0d0) + (-1.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around inf 99.3%

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

3. Taylor expanded in y around 0 61.3%

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

   1. associate-*r/ 61.3%

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

   2. associate-*r* 61.3%

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

   3. sub-neg 61.3%

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

   4. metadata-eval 61.3%

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

   5. distribute-lft-out 61.3%

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

   6. *-commutative 61.3%

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

   7. sub-neg 61.3%

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

   8. metadata-eval 61.3%

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

5. Simplified 61.3%

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

6. Final simplification 61.3%

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

Alternative 27: 40.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{6} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (- (cos x) (cos y))
    (* (* (sqrt 2.0) (sin x)) (- (sin y) (/ (sin x) 16.0)))))
  6.0))

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / 6.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * ((sqrt(2.0d0) * sin(x)) * (sin(y) - (sin(x) / 16.0d0))))) / 6.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * ((Math.sqrt(2.0) * Math.sin(x)) * (Math.sin(y) - (Math.sin(x) / 16.0))))) / 6.0;
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * ((math.sqrt(2.0) * math.sin(x)) * (math.sin(y) - (math.sin(x) / 16.0))))) / 6.0

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(Float64(sqrt(2.0) * sin(x)) * Float64(sin(y) - Float64(sin(x) / 16.0))))) / 6.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * ((sqrt(2.0) * sin(x)) * (sin(y) - (sin(x) / 16.0))))) / 6.0;
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in y around 0 65.2%

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

3. Taylor expanded in y around 0 61.5%

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

   1. distribute-rgt-in 61.5%

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

   2. metadata-eval 61.5%

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

   3. distribute-lft-out 61.5%

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

   4. *-commutative 61.5%

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

   5. sub-neg 61.5%

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

   6. metadata-eval 61.5%

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

5. Simplified 61.5%

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

6. Taylor expanded in x around 0 43.1%

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

7. Final simplification 43.1%

   \[\leadsto \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sqrt{2} \cdot \sin x\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)}{6} \]

Alternative 28: 32.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+ 2.0 (* (pow y 4.0) (* (sqrt 2.0) -0.03125)))
  (*
   3.0
   (+ 1.0 (* 0.5 (+ (- 3.0 (sqrt 5.0)) (* (cos x) (+ (sqrt 5.0) -1.0))))))))

C

double code(double x, double y) {
	return (2.0 + (pow(y, 4.0) * (sqrt(2.0) * -0.03125))) / (3.0 * (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * (sqrt(5.0) + -1.0))))));
}

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)))) / (3.0d0 * (1.0d0 + (0.5d0 * ((3.0d0 - sqrt(5.0d0)) + (cos(x) * (sqrt(5.0d0) + (-1.0d0)))))))
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))) / (3.0 * (1.0 + (0.5 * ((3.0 - Math.sqrt(5.0)) + (Math.cos(x) * (Math.sqrt(5.0) + -1.0))))));
}

Python

def code(x, y):
	return (2.0 + (math.pow(y, 4.0) * (math.sqrt(2.0) * -0.03125))) / (3.0 * (1.0 + (0.5 * ((3.0 - math.sqrt(5.0)) + (math.cos(x) * (math.sqrt(5.0) + -1.0))))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64((y ^ 4.0) * Float64(sqrt(2.0) * -0.03125))) / Float64(3.0 * Float64(1.0 + Float64(0.5 * Float64(Float64(3.0 - sqrt(5.0)) + Float64(cos(x) * Float64(sqrt(5.0) + -1.0)))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((y ^ 4.0) * (sqrt(2.0) * -0.03125))) / (3.0 * (1.0 + (0.5 * ((3.0 - sqrt(5.0)) + (cos(x) * (sqrt(5.0) + -1.0))))));
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] / N[(3.0 * N[(1.0 + N[(0.5 * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in x around 0 62.8%

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

   1. associate-*r* 62.8%

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

4. Simplified 62.8%

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

5. Taylor expanded in y around 0 34.9%

   \[\leadsto \frac{2 + \color{blue}{-0.03125 \cdot \left(\sqrt{2} \cdot {y}^{4}\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. *-commutative 34.9%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot {y}^{4}\right) \cdot -0.03125}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative 34.9%

      \[\leadsto \frac{2 + \color{blue}{\left({y}^{4} \cdot \sqrt{2}\right)} \cdot -0.03125}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* 34.9%

      \[\leadsto \frac{2 + \color{blue}{{y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

7. Simplified 34.9%

   \[\leadsto \frac{2 + \color{blue}{{y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0 34.9%

   \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
9. Step-by-step derivation

   1. distribute-lft-out 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)}\right)} \]

   2. *-commutative 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)} + \left(3 - \sqrt{5}\right)\right)\right)} \]

   3. sub-neg 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)\right)} \]

   4. metadata-eval 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)\right)} \]

10. Simplified 34.9%

    \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \color{blue}{\left(1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)\right)}} \]

11. Final simplification 34.9%

    \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\left(3 - \sqrt{5}\right) + \cos x \cdot \left(\sqrt{5} + -1\right)\right)\right)} \]

Alternative 29: 29.9% 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%

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

2. Taylor expanded in x around 0 62.8%

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

   1. associate-*r* 62.8%

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

4. Simplified 62.8%

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

5. Taylor expanded in y around 0 34.9%

   \[\leadsto \frac{2 + \color{blue}{-0.03125 \cdot \left(\sqrt{2} \cdot {y}^{4}\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]
6. Step-by-step derivation

   1. *-commutative 34.9%

      \[\leadsto \frac{2 + \color{blue}{\left(\sqrt{2} \cdot {y}^{4}\right) \cdot -0.03125}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   2. *-commutative 34.9%

      \[\leadsto \frac{2 + \color{blue}{\left({y}^{4} \cdot \sqrt{2}\right)} \cdot -0.03125}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

   3. associate-*l* 34.9%

      \[\leadsto \frac{2 + \color{blue}{{y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

7. Simplified 34.9%

   \[\leadsto \frac{2 + \color{blue}{{y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)} \]

8. Taylor expanded in y around 0 34.9%

   \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \color{blue}{\left(1 + \left(0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x\right) + 0.5 \cdot \left(3 - \sqrt{5}\right)\right)\right)}} \]
9. Step-by-step derivation

   1. distribute-lft-out 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + \color{blue}{0.5 \cdot \left(\left(\sqrt{5} - 1\right) \cdot \cos x + \left(3 - \sqrt{5}\right)\right)}\right)} \]

   2. *-commutative 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\color{blue}{\cos x \cdot \left(\sqrt{5} - 1\right)} + \left(3 - \sqrt{5}\right)\right)\right)} \]

   3. sub-neg 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot \color{blue}{\left(\sqrt{5} + \left(-1\right)\right)} + \left(3 - \sqrt{5}\right)\right)\right)} \]

   4. metadata-eval 34.9%

      \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \left(1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + \color{blue}{-1}\right) + \left(3 - \sqrt{5}\right)\right)\right)} \]

10. Simplified 34.9%

    \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{3 \cdot \color{blue}{\left(1 + 0.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \left(3 - \sqrt{5}\right)\right)\right)}} \]

11. Taylor expanded in x around 0 32.9%

    \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{\color{blue}{6}} \]

12. Final simplification 32.9%

    \[\leadsto \frac{2 + {y}^{4} \cdot \left(\sqrt{2} \cdot -0.03125\right)}{6} \]

Reproduce

herbie shell --seed 2023196 
(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))))))