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

Percentage Accurate: 99.3% → 99.4%

Time: 45.4s

Alternatives: 24

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{1.5 + \sqrt{11.25}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin y) (* (sin x) -0.0625))
    (* (+ (sin x) (* (sin y) -0.0625)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (fma
    (cos y)
    (* (- 3.0 (sqrt 5.0)) 1.5)
    (* (cos x) (/ 9.0 (+ 1.5 (sqrt 11.25))))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (sin(y) * -0.0625)) * (cos(x) - cos(y)))), 2.0) / (3.0 + fma(cos(y), ((3.0 - sqrt(5.0)) * 1.5), (cos(x) * (9.0 / (1.5 + sqrt(11.25))))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + fma(cos(y), Float64(Float64(3.0 - sqrt(5.0)) * 1.5), Float64(cos(x) * Float64(9.0 / Float64(1.5 + sqrt(11.25)))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(9.0 / N[(1.5 + N[Sqrt[11.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.4%

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

   4. sub-neg 99.4%

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

   5. metadata-eval 99.4%

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

   6. distribute-lft-in 99.4%

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

   7. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. flip-+ 99.2%

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

   2. swap-sqr 99.2%

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

   3. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{\color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right) - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   4. rem-square-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{2.25 \cdot \color{blue}{5} - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{\color{blue}{11.25} - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{11.25 - \color{blue}{2.25}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. add-sqr-sqrt 99.4%

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

   10. sqrt-unprod 99.4%

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

   11. swap-sqr 99.4%

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

   12. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{\color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)} + \left(--1.5\right)}\right)} \]

   13. rem-square-sqrt 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{2.25 \cdot \color{blue}{5}} + \left(--1.5\right)}\right)} \]

   14. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{\color{blue}{11.25}} + \left(--1.5\right)}\right)} \]

   15. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{11.25} + \color{blue}{1.5}}\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \color{blue}{\frac{9}{\sqrt{11.25} + 1.5}}\right)} \]

10. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{1.5 + \sqrt{11.25}}\right)} \]
11. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 9 \cdot \frac{\cos x}{1.5 + \sqrt{11.25}}\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin y) (* (sin x) -0.0625))
    (* (+ (sin x) (* (sin y) -0.0625)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (+
    (* 1.5 (* (cos y) (- 3.0 (sqrt 5.0))))
    (* 9.0 (/ (cos x) (+ 1.5 (sqrt 11.25))))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (sin(y) * -0.0625)) * (cos(x) - cos(y)))), 2.0) / (3.0 + ((1.5 * (cos(y) * (3.0 - sqrt(5.0)))) + (9.0 * (cos(x) / (1.5 + sqrt(11.25))))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(1.5 * Float64(cos(y) * Float64(3.0 - sqrt(5.0)))) + Float64(9.0 * Float64(cos(x) / Float64(1.5 + sqrt(11.25)))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(1.5 * N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[(N[Cos[x], $MachinePrecision] / N[(1.5 + N[Sqrt[11.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.4%

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

   4. sub-neg 99.4%

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

   5. metadata-eval 99.4%

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

   6. distribute-lft-in 99.4%

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

   7. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. flip-+ 99.2%

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

   2. swap-sqr 99.2%

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

   3. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{\color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right) - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   4. rem-square-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{2.25 \cdot \color{blue}{5} - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{\color{blue}{11.25} - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{11.25 - \color{blue}{2.25}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. add-sqr-sqrt 99.4%

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

   10. sqrt-unprod 99.4%

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

   11. swap-sqr 99.4%

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

   12. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{\color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)} + \left(--1.5\right)}\right)} \]

   13. rem-square-sqrt 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{2.25 \cdot \color{blue}{5}} + \left(--1.5\right)}\right)} \]

   14. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{\color{blue}{11.25}} + \left(--1.5\right)}\right)} \]

   15. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{11.25} + \color{blue}{1.5}}\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \color{blue}{\frac{9}{\sqrt{11.25} + 1.5}}\right)} \]

10. Taylor expanded in y around inf 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 9 \cdot \frac{\cos x}{1.5 + \sqrt{11.25}}\right)}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(1.5 \cdot \left(\cos y \cdot \left(3 - \sqrt{5}\right)\right) + 9 \cdot \frac{\cos x}{1.5 + \sqrt{11.25}}\right)} \]
12. Add Preprocessing

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \left(\sqrt{11.25} + -1.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin y) (* (sin x) -0.0625))
    (* (+ (sin x) (* (sin y) -0.0625)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (+
    (* (cos y) (* (- 3.0 (sqrt 5.0)) 1.5))
    (* (cos x) (+ (sqrt 11.25) -1.5))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(y) + (sin(x) * -0.0625)) * ((sin(x) + (sin(y) * -0.0625)) * (cos(x) - cos(y)))), 2.0) / (3.0 + ((cos(y) * ((3.0 - sqrt(5.0)) * 1.5)) + (cos(x) * (sqrt(11.25) + -1.5))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(y) + Float64(sin(x) * -0.0625)) * Float64(Float64(sin(x) + Float64(sin(y) * -0.0625)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) * 1.5)) + Float64(cos(x) * Float64(sqrt(11.25) + -1.5)))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[11.25], $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around inf 99.3%

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

   1. associate-*r* 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.4%

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

   4. sub-neg 99.4%

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

   5. metadata-eval 99.4%

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

   6. distribute-lft-in 99.4%

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

   7. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. flip-+ 99.2%

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

   2. swap-sqr 99.2%

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

   3. metadata-eval 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{\color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right) - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   4. rem-square-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{2.25 \cdot \color{blue}{5} - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{\color{blue}{11.25} - -1.5 \cdot -1.5}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{11.25 - \color{blue}{2.25}}{1.5 \cdot \sqrt{5} - -1.5}\right)} \]

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. add-sqr-sqrt 99.4%

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

   10. sqrt-unprod 99.4%

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

   11. swap-sqr 99.4%

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

   12. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{\color{blue}{2.25} \cdot \left(\sqrt{5} \cdot \sqrt{5}\right)} + \left(--1.5\right)}\right)} \]

   13. rem-square-sqrt 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{2.25 \cdot \color{blue}{5}} + \left(--1.5\right)}\right)} \]

   14. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{\color{blue}{11.25}} + \left(--1.5\right)}\right)} \]

   15. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \frac{9}{\sqrt{11.25} + \color{blue}{1.5}}\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \left(3 - \sqrt{5}\right) \cdot 1.5, \cos x \cdot \color{blue}{\frac{9}{\sqrt{11.25} + 1.5}}\right)} \]
10. Step-by-step derivation

    1. fma-undefine 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \frac{9}{\sqrt{11.25} + 1.5}\right)}} \]

    2. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \frac{\color{blue}{11.25 - 2.25}}{\sqrt{11.25} + 1.5}\right)} \]

    3. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \frac{11.25 - \color{blue}{1.5 \cdot 1.5}}{\sqrt{11.25} + 1.5}\right)} \]

    4. rem-square-sqrt 99.2%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \frac{\color{blue}{\sqrt{11.25} \cdot \sqrt{11.25}} - 1.5 \cdot 1.5}{\sqrt{11.25} + 1.5}\right)} \]

    5. flip-- 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \color{blue}{\left(\sqrt{11.25} - 1.5\right)}\right)} \]

    6. sub-neg 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \color{blue}{\left(\sqrt{11.25} + \left(-1.5\right)\right)}\right)} \]

    7. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \left(\sqrt{11.25} + \color{blue}{-1.5}\right)\right)} \]

11. Applied egg-rr 99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \left(\sqrt{11.25} + -1.5\right)\right)}} \]

12. Final simplification 99.3%

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin y + \sin x \cdot -0.0625\right) \cdot \left(\left(\sin x + \sin y \cdot -0.0625\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \left(\left(3 - \sqrt{5}\right) \cdot 1.5\right) + \cos x \cdot \left(\sqrt{11.25} + -1.5\right)\right)} \]
13. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. flip-- 61.4%

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

   2. metadata-eval 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

   3. pow1/2 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

   4. pow1/2 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

   5. pow-prod-up 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

   6. metadata-eval 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

   7. metadata-eval 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

   8. metadata-eval 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

4. Applied egg-rr 99.3%

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

   1. +-commutative 61.4%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (- (cos x) (cos y))
    (*
     (sqrt 2.0)
     (* (- (sin x) (/ (sin y) 16.0)) (- (sin y) (/ (sin x) 16.0))))))
  (*
   3.0
   (+
    1.0
    (+ (* (cos y) (- 1.5 (sqrt 1.25))) (* (cos x) (+ (sqrt 1.25) -0.5)))))))

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(y) * (1.5 - sqrt(1.25))) + (cos(x) * (sqrt(1.25) + -0.5)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 + ((cos(x) - cos(y)) * (sqrt(2.0d0) * ((sin(x) - (sin(y) / 16.0d0)) * (sin(y) - (sin(x) / 16.0d0)))))) / (3.0d0 * (1.0d0 + ((cos(y) * (1.5d0 - sqrt(1.25d0))) + (cos(x) * (sqrt(1.25d0) + (-0.5d0))))))
end function

Java

public static double code(double x, double y) {
	return (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.sqrt(2.0) * ((Math.sin(x) - (Math.sin(y) / 16.0)) * (Math.sin(y) - (Math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((Math.cos(y) * (1.5 - Math.sqrt(1.25))) + (Math.cos(x) * (Math.sqrt(1.25) + -0.5)))));
}

Python

def code(x, y):
	return (2.0 + ((math.cos(x) - math.cos(y)) * (math.sqrt(2.0) * ((math.sin(x) - (math.sin(y) / 16.0)) * (math.sin(y) - (math.sin(x) / 16.0)))))) / (3.0 * (1.0 + ((math.cos(y) * (1.5 - math.sqrt(1.25))) + (math.cos(x) * (math.sqrt(1.25) + -0.5)))))

Julia

function code(x, y)
	return Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(Float64(sin(x) - Float64(sin(y) / 16.0)) * Float64(sin(y) - Float64(sin(x) / 16.0)))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(cos(y) * Float64(1.5 - sqrt(1.25))) + Float64(cos(x) * Float64(sqrt(1.25) + -0.5))))))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * ((sin(x) - (sin(y) / 16.0)) * (sin(y) - (sin(x) / 16.0)))))) / (3.0 * (1.0 + ((cos(y) * (1.5 - sqrt(1.25))) + (cos(x) * (sqrt(1.25) + -0.5)))));
end

Wolfram

code[x_, y_] := N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[1.25], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

   5. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. fma-define 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. fma-neg 99.3%

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

   6. metadata-eval 99.3%

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

   7. div-inv 99.3%

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

   8. metadata-eval 99.3%

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

6. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. fma-undefine 99.3%

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

   3. add-sqr-sqrt 98.6%

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

   4. sqrt-unprod 99.3%

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

   5. swap-sqr 99.3%

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

   6. pow1/2 99.3%

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

   7. pow1/2 99.3%

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

   8. pow-prod-up 99.3%

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

   9. metadata-eval 99.3%

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

   10. metadata-eval 99.3%

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

   11. metadata-eval 99.3%

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

   12. metadata-eval 99.3%

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

   13. add-sqr-sqrt 99.1%

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

   14. sqrt-unprod 99.3%

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

   15. swap-sqr 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

   \[\leadsto \frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)}{3 \cdot \left(1 + \left(\cos y \cdot \left(1.5 - \sqrt{1.25}\right) + \cos x \cdot \left(\sqrt{1.25} + -0.5\right)\right)\right)} \]
10. Add Preprocessing

Alternative 7: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := \cos x - \cos y\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := 2 + t\_1 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ t_4 := 3 \cdot \left(t\_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ \mathbf{if}\;x \leq -0.0225:\\ \;\;\;\;\frac{t\_3}{3 \cdot \left(t\_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + t\_1 \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_1 (- (cos x) (cos y)))
        (t_2 (- (sin y) (/ (sin x) 16.0)))
        (t_3 (+ 2.0 (* t_1 (* t_2 (* (sqrt 2.0) (sin x))))))
        (t_4 (* 3.0 (+ t_0 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))
   (if (<= x -0.0225)
     (/ t_3 (* 3.0 (+ t_0 (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0)))))
     (if (<= x 1.9e-8)
       (/ (+ 2.0 (* t_1 (* t_2 (* (sqrt 2.0) (+ x (* (sin y) -0.0625)))))) t_4)
       (/ t_3 t_4)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0));
	double t_1 = cos(x) - cos(y);
	double t_2 = sin(y) - (sin(x) / 16.0);
	double t_3 = 2.0 + (t_1 * (t_2 * (sqrt(2.0) * sin(x))));
	double t_4 = 3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	double tmp;
	if (x <= -0.0225) {
		tmp = t_3 / (3.0 * (t_0 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	} else if (x <= 1.9e-8) {
		tmp = (2.0 + (t_1 * (t_2 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_4;
	} else {
		tmp = t_3 / 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 = 1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))
    t_1 = cos(x) - cos(y)
    t_2 = sin(y) - (sin(x) / 16.0d0)
    t_3 = 2.0d0 + (t_1 * (t_2 * (sqrt(2.0d0) * sin(x))))
    t_4 = 3.0d0 * (t_0 + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0)))
    if (x <= (-0.0225d0)) then
        tmp = t_3 / (3.0d0 * (t_0 + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    else if (x <= 1.9d-8) then
        tmp = (2.0d0 + (t_1 * (t_2 * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))))) / t_4
    else
        tmp = t_3 / t_4
    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 = Math.cos(x) - Math.cos(y);
	double t_2 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_3 = 2.0 + (t_1 * (t_2 * (Math.sqrt(2.0) * Math.sin(x))));
	double t_4 = 3.0 * (t_0 + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0)));
	double tmp;
	if (x <= -0.0225) {
		tmp = t_3 / (3.0 * (t_0 + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0))));
	} else if (x <= 1.9e-8) {
		tmp = (2.0 + (t_1 * (t_2 * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))))) / t_4;
	} else {
		tmp = t_3 / t_4;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))
	t_1 = math.cos(x) - math.cos(y)
	t_2 = math.sin(y) - (math.sin(x) / 16.0)
	t_3 = 2.0 + (t_1 * (t_2 * (math.sqrt(2.0) * math.sin(x))))
	t_4 = 3.0 * (t_0 + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0)))
	tmp = 0
	if x <= -0.0225:
		tmp = t_3 / (3.0 * (t_0 + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0))))
	elif x <= 1.9e-8:
		tmp = (2.0 + (t_1 * (t_2 * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))))) / t_4
	else:
		tmp = t_3 / t_4
	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(cos(x) - cos(y))
	t_2 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_3 = Float64(2.0 + Float64(t_1 * Float64(t_2 * Float64(sqrt(2.0) * sin(x)))))
	t_4 = Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0))))
	tmp = 0.0
	if (x <= -0.0225)
		tmp = Float64(t_3 / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	elseif (x <= 1.9e-8)
		tmp = Float64(Float64(2.0 + Float64(t_1 * Float64(t_2 * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))))) / t_4);
	else
		tmp = Float64(t_3 / t_4);
	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 = cos(x) - cos(y);
	t_2 = sin(y) - (sin(x) / 16.0);
	t_3 = 2.0 + (t_1 * (t_2 * (sqrt(2.0) * sin(x))));
	t_4 = 3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	tmp = 0.0;
	if (x <= -0.0225)
		tmp = t_3 / (3.0 * (t_0 + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	elseif (x <= 1.9e-8)
		tmp = (2.0 + (t_1 * (t_2 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_4;
	else
		tmp = t_3 / t_4;
	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[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[x, -0.0225], N[(t$95$3 / 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]), $MachinePrecision], If[LessEqual[x, 1.9e-8], N[(N[(2.0 + N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(t$95$3 / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.022499999999999999

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 66.2%

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

   if -0.022499999999999999 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. distribute-rgt-out 99.5%

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

   5. Simplified 99.5%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   7. Applied egg-rr 99.6%

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

      1. +-commutative 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   9. Simplified 99.6%

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

   if 1.90000000000000014e-8 < x

   1. Initial program 99.0%

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

      1. flip-- 22.6%

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

      2. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   4. Applied egg-rr 99.0%

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

      1. +-commutative 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   6. Simplified 99.0%

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

   7. Taylor expanded in y around 0 69.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0225:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t\_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \cos x - \cos y\\ t_3 := \sin y - \frac{\sin x}{16}\\ t_4 := 2 + t\_2 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ \mathbf{if}\;x \leq -0.0275:\\ \;\;\;\;\frac{t\_4}{t\_1}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + t\_2 \cdot \left(t\_3 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{3 \cdot \left(t\_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

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 (- (cos x) (cos y)))
        (t_3 (- (sin y) (/ (sin x) 16.0)))
        (t_4 (+ 2.0 (* t_2 (* t_3 (* (sqrt 2.0) (sin x)))))))
   (if (<= x -0.0275)
     (/ t_4 t_1)
     (if (<= x 1.9e-8)
       (/ (+ 2.0 (* t_2 (* t_3 (* (sqrt 2.0) (+ x (* (sin y) -0.0625)))))) t_1)
       (/
        t_4
        (* 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 = cos(x) - cos(y);
	double t_3 = sin(y) - (sin(x) / 16.0);
	double t_4 = 2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))));
	double tmp;
	if (x <= -0.0275) {
		tmp = t_4 / t_1;
	} else if (x <= 1.9e-8) {
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_1;
	} else {
		tmp = t_4 / (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) :: t_3
    real(8) :: t_4
    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 = cos(x) - cos(y)
    t_3 = sin(y) - (sin(x) / 16.0d0)
    t_4 = 2.0d0 + (t_2 * (t_3 * (sqrt(2.0d0) * sin(x))))
    if (x <= (-0.0275d0)) then
        tmp = t_4 / t_1
    else if (x <= 1.9d-8) then
        tmp = (2.0d0 + (t_2 * (t_3 * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))))) / t_1
    else
        tmp = t_4 / (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 = Math.cos(x) - Math.cos(y);
	double t_3 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_4 = 2.0 + (t_2 * (t_3 * (Math.sqrt(2.0) * Math.sin(x))));
	double tmp;
	if (x <= -0.0275) {
		tmp = t_4 / t_1;
	} else if (x <= 1.9e-8) {
		tmp = (2.0 + (t_2 * (t_3 * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))))) / t_1;
	} else {
		tmp = t_4 / (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 = math.cos(x) - math.cos(y)
	t_3 = math.sin(y) - (math.sin(x) / 16.0)
	t_4 = 2.0 + (t_2 * (t_3 * (math.sqrt(2.0) * math.sin(x))))
	tmp = 0
	if x <= -0.0275:
		tmp = t_4 / t_1
	elif x <= 1.9e-8:
		tmp = (2.0 + (t_2 * (t_3 * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))))) / t_1
	else:
		tmp = t_4 / (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(cos(x) - cos(y))
	t_3 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_4 = Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sqrt(2.0) * sin(x)))))
	tmp = 0.0
	if (x <= -0.0275)
		tmp = Float64(t_4 / t_1);
	elseif (x <= 1.9e-8)
		tmp = Float64(Float64(2.0 + Float64(t_2 * Float64(t_3 * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))))) / t_1);
	else
		tmp = Float64(t_4 / 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 = cos(x) - cos(y);
	t_3 = sin(y) - (sin(x) / 16.0);
	t_4 = 2.0 + (t_2 * (t_3 * (sqrt(2.0) * sin(x))));
	tmp = 0.0;
	if (x <= -0.0275)
		tmp = t_4 / t_1;
	elseif (x <= 1.9e-8)
		tmp = (2.0 + (t_2 * (t_3 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))))) / t_1;
	else
		tmp = t_4 / (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[Cos[x], $MachinePrecision] - N[Cos[y], $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[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0275], N[(t$95$4 / t$95$1), $MachinePrecision], If[LessEqual[x, 1.9e-8], N[(N[(2.0 + N[(t$95$2 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$4 / 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 x < -0.0275000000000000001

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 66.2%

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

   if -0.0275000000000000001 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. distribute-rgt-out 99.5%

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

   5. Simplified 99.5%

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

   if 1.90000000000000014e-8 < x

   1. Initial program 99.0%

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

      1. flip-- 22.6%

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

      2. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   4. Applied egg-rr 99.0%

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

      1. +-commutative 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   6. Simplified 99.0%

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

   7. Taylor expanded in y around 0 69.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0275:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := 3 \cdot \left(t\_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ t_2 := \sin y - \frac{\sin x}{16}\\ t_3 := 2 + \left(\cos x - \cos y\right) \cdot \left(t\_2 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)\\ \mathbf{if}\;x \leq -0.0053:\\ \;\;\;\;\frac{t\_3}{t\_1}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(t\_2 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{3 \cdot \left(t\_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

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 (- (sin y) (/ (sin x) 16.0)))
        (t_3 (+ 2.0 (* (- (cos x) (cos y)) (* t_2 (* (sqrt 2.0) (sin x)))))))
   (if (<= x -0.0053)
     (/ t_3 t_1)
     (if (<= x 1.9e-8)
       (/
        (+
         2.0
         (* (* t_2 (* (sqrt 2.0) (+ x (* (sin y) -0.0625)))) (- 1.0 (cos y))))
        t_1)
       (/
        t_3
        (* 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 = sin(y) - (sin(x) / 16.0);
	double t_3 = 2.0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0) * sin(x))));
	double tmp;
	if (x <= -0.0053) {
		tmp = t_3 / t_1;
	} else if (x <= 1.9e-8) {
		tmp = (2.0 + ((t_2 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))) * (1.0 - cos(y)))) / t_1;
	} else {
		tmp = t_3 / (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) :: t_3
    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 = sin(y) - (sin(x) / 16.0d0)
    t_3 = 2.0d0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0d0) * sin(x))))
    if (x <= (-0.0053d0)) then
        tmp = t_3 / t_1
    else if (x <= 1.9d-8) then
        tmp = (2.0d0 + ((t_2 * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))) * (1.0d0 - cos(y)))) / t_1
    else
        tmp = t_3 / (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 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_3 = 2.0 + ((Math.cos(x) - Math.cos(y)) * (t_2 * (Math.sqrt(2.0) * Math.sin(x))));
	double tmp;
	if (x <= -0.0053) {
		tmp = t_3 / t_1;
	} else if (x <= 1.9e-8) {
		tmp = (2.0 + ((t_2 * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))) * (1.0 - Math.cos(y)))) / t_1;
	} else {
		tmp = t_3 / (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 = math.sin(y) - (math.sin(x) / 16.0)
	t_3 = 2.0 + ((math.cos(x) - math.cos(y)) * (t_2 * (math.sqrt(2.0) * math.sin(x))))
	tmp = 0
	if x <= -0.0053:
		tmp = t_3 / t_1
	elif x <= 1.9e-8:
		tmp = (2.0 + ((t_2 * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))) * (1.0 - math.cos(y)))) / t_1
	else:
		tmp = t_3 / (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(sin(y) - Float64(sin(x) / 16.0))
	t_3 = Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_2 * Float64(sqrt(2.0) * sin(x)))))
	tmp = 0.0
	if (x <= -0.0053)
		tmp = Float64(t_3 / t_1);
	elseif (x <= 1.9e-8)
		tmp = Float64(Float64(2.0 + Float64(Float64(t_2 * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))) * Float64(1.0 - cos(y)))) / t_1);
	else
		tmp = Float64(t_3 / 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 = sin(y) - (sin(x) / 16.0);
	t_3 = 2.0 + ((cos(x) - cos(y)) * (t_2 * (sqrt(2.0) * sin(x))));
	tmp = 0.0;
	if (x <= -0.0053)
		tmp = t_3 / t_1;
	elseif (x <= 1.9e-8)
		tmp = (2.0 + ((t_2 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))) * (1.0 - cos(y)))) / t_1;
	else
		tmp = t_3 / (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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0053], N[(t$95$3 / t$95$1), $MachinePrecision], If[LessEqual[x, 1.9e-8], N[(N[(2.0 + N[(N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$3 / 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 x < -0.00530000000000000002

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 66.2%

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

   if -0.00530000000000000002 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. distribute-rgt-out 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.4%

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

   if 1.90000000000000014e-8 < x

   1. Initial program 99.0%

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

      1. flip-- 22.6%

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

      2. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   4. Applied egg-rr 99.0%

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

      1. +-commutative 22.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   6. Simplified 99.0%

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

   7. Taylor expanded in y around 0 69.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0053:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0057 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t\_0 \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sin y) (/ (sin x) 16.0)))
        (t_1
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (- 3.0 (sqrt 5.0)) 2.0))))))
   (if (or (<= x -0.0057) (not (<= x 1.9e-8)))
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* t_0 (* (sqrt 2.0) (sin x))))) t_1)
     (/
      (+
       2.0
       (* (* t_0 (* (sqrt 2.0) (+ x (* (sin y) -0.0625)))) (- 1.0 (cos y))))
      t_1))))

C

double code(double x, double y) {
	double t_0 = sin(y) - (sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.0057) || !(x <= 1.9e-8)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0) * sin(x))))) / t_1;
	} else {
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))) * (1.0 - cos(y)))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(y) - (sin(x) / 16.0d0)
    t_1 = 3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0)))
    if ((x <= (-0.0057d0)) .or. (.not. (x <= 1.9d-8))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0d0) * sin(x))))) / t_1
    else
        tmp = (2.0d0 + ((t_0 * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))) * (1.0d0 - cos(y)))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) - (Math.sin(x) / 16.0);
	double t_1 = 3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((3.0 - Math.sqrt(5.0)) / 2.0)));
	double tmp;
	if ((x <= -0.0057) || !(x <= 1.9e-8)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (t_0 * (Math.sqrt(2.0) * Math.sin(x))))) / t_1;
	} else {
		tmp = (2.0 + ((t_0 * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))) * (1.0 - Math.cos(y)))) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) - (math.sin(x) / 16.0)
	t_1 = 3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((3.0 - math.sqrt(5.0)) / 2.0)))
	tmp = 0
	if (x <= -0.0057) or not (x <= 1.9e-8):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (t_0 * (math.sqrt(2.0) * math.sin(x))))) / t_1
	else:
		tmp = (2.0 + ((t_0 * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))) * (1.0 - math.cos(y)))) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_1 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0))))
	tmp = 0.0
	if ((x <= -0.0057) || !(x <= 1.9e-8))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_0 * Float64(sqrt(2.0) * sin(x))))) / t_1);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(t_0 * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))) * Float64(1.0 - cos(y)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) - (sin(x) / 16.0);
	t_1 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0)));
	tmp = 0.0;
	if ((x <= -0.0057) || ~((x <= 1.9e-8)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_0 * (sqrt(2.0) * sin(x))))) / t_1;
	else
		tmp = (2.0 + ((t_0 * (sqrt(2.0) * (x + (sin(y) * -0.0625)))) * (1.0 - cos(y)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -0.0057], N[Not[LessEqual[x, 1.9e-8]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(2.0 + N[(N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0057000000000000002 or 1.90000000000000014e-8 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 68.0%

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

   if -0.0057000000000000002 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. distribute-rgt-out 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0057 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.00245 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.00245) (not (<= x 1.9e-8)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+
       2.0
       (*
        (*
         (- (sin y) (/ (sin x) 16.0))
         (* (sqrt 2.0) (+ x (* (sin y) -0.0625))))
        (- 1.0 (cos 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) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00245) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (x + (sin(y) * -0.0625)))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-0.00245d0)) .or. (.not. (x <= 1.9d-8))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + (((sin(y) - (sin(x) / 16.0d0)) * (sqrt(2.0d0) * (x + (sin(y) * (-0.0625d0))))) * (1.0d0 - cos(y)))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((3.0d0 - sqrt(5.0d0)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00245) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sqrt(2.0) * (x + (Math.sin(y) * -0.0625)))) * (1.0 - Math.cos(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))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -0.00245) or not (x <= 1.9e-8):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (((math.sin(y) - (math.sin(x) / 16.0)) * (math.sqrt(2.0) * (x + (math.sin(y) * -0.0625)))) * (1.0 - math.cos(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))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -0.00245) || !(x <= 1.9e-8))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(x + Float64(sin(y) * -0.0625)))) * Float64(1.0 - cos(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
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -0.00245) || ~((x <= 1.9e-8)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (x + (sin(y) * -0.0625)))) * (1.0 - cos(y)))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((3.0 - sqrt(5.0)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -0.00245], N[Not[LessEqual[x, 1.9e-8]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(x + N[(N[Sin[y], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0024499999999999999 or 1.90000000000000014e-8 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in y around 0 64.8%

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

   9. Taylor expanded in x around inf 64.9%

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

   if -0.0024499999999999999 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-*r* 99.5%

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

      3. distribute-rgt-out 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00245 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(x + \sin y \cdot -0.0625\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)\\ t_1 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{t\_0}\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, -\sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          3.0
          (+
           (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
           (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
        (t_1 (pow (sin y) 2.0)))
   (if (<= y -6.8e-6)
     (/ (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) (- 1.0 (cos y)))))) t_0)
     (if (<= y 0.0007)
       (/
        (*
         0.3333333333333333
         (fma
          -0.0625
          (* (+ (cos x) -1.0) (* (sqrt 2.0) (pow (sin x) 2.0)))
          2.0))
        (+ 1.0 (+ 1.5 (fma (cos x) (+ (sqrt 1.25) -0.5) (- (sqrt 1.25))))))
       (/
        (+ 2.0 (* (- (cos x) (cos y)) (* t_1 (* (sqrt 2.0) -0.0625))))
        t_0)))))

C

double code(double x, double y) {
	double t_0 = 3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0)));
	double t_1 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -6.8e-6) {
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (1.0 - cos(y)))))) / t_0;
	} else if (y <= 0.0007) {
		tmp = (0.3333333333333333 * fma(-0.0625, ((cos(x) + -1.0) * (sqrt(2.0) * pow(sin(x), 2.0))), 2.0)) / (1.0 + (1.5 + fma(cos(x), (sqrt(1.25) + -0.5), -sqrt(1.25))));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_1 * (sqrt(2.0) * -0.0625)))) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0))))
	t_1 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -6.8e-6)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / t_0);
	elseif (y <= 0.0007)
		tmp = Float64(Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * (sin(x) ^ 2.0))), 2.0)) / Float64(1.0 + Float64(1.5 + fma(cos(x), Float64(sqrt(1.25) + -0.5), Float64(-sqrt(1.25))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_1 * Float64(sqrt(2.0) * -0.0625)))) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -6.8e-6], N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 0.0007], N[(N[(0.3333333333333333 * 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] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.5 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[1.25], $MachinePrecision] + -0.5), $MachinePrecision] + (-N[Sqrt[1.25], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.80000000000000012e-6

   1. Initial program 99.2%

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

      1. flip-- 68.5%

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

      2. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   4. Applied egg-rr 99.3%

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

      1. +-commutative 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 69.8%

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

   if -6.80000000000000012e-6 < y < 6.99999999999999993e-4

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-define 98.6%

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

      4. *-commutative 98.6%

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

      5. *-commutative 98.6%

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

      6. associate-*l* 98.6%

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

      7. sub-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. fma-undefine 98.6%

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

      2. fma-undefine 98.6%

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

      3. metadata-eval 98.6%

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

      4. distribute-rgt-neg-in 98.6%

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

      5. associate-+r+ 98.6%

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

   9. Applied egg-rr 98.6%

      \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(-\sqrt{1.25}\right)\right) + 1.5\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 98.6%

          \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(1.5 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(-\sqrt{1.25}\right)\right)\right)}} \]

       2. fma-define 98.6%

          \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, -\sqrt{1.25}\right)}\right)} \]

       3. +-commutative 98.6%

          \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \mathsf{fma}\left(\cos x, \color{blue}{-0.5 + \sqrt{1.25}}, -\sqrt{1.25}\right)\right)} \]

   11. Simplified 98.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(1.5 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{1.25}, -\sqrt{1.25}\right)\right)}} \]

   if 6.99999999999999993e-4 < y

   1. Initial program 99.1%

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

      1. flip-- 61.9%

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

      2. metadata-eval 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   4. Applied egg-rr 99.1%

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

      1. +-commutative 61.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   6. Simplified 99.1%

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

   7. Taylor expanded in x around 0 63.6%

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

      1. *-commutative 63.6%

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

      2. associate-*l* 63.6%

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

   9. Simplified 63.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, -\sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot -0.0625\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\\ t_1 := {\sin y}^{2}\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(t\_0 + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, -\sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot t\_1\right)\right)}{3 \cdot \left(t\_0 + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0))))
        (t_1 (pow (sin y) 2.0)))
   (if (<= y -2.9e-5)
     (/
      (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) (- 1.0 (cos y))))))
      (* 3.0 (+ t_0 (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0)))))
     (if (<= y 0.0007)
       (/
        (*
         0.3333333333333333
         (fma
          -0.0625
          (* (+ (cos x) -1.0) (* (sqrt 2.0) (pow (sin x) 2.0)))
          2.0))
        (+ 1.0 (+ 1.5 (fma (cos x) (+ (sqrt 1.25) -0.5) (- (sqrt 1.25))))))
       (/
        (+ 2.0 (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 t_1))))
        (* 3.0 (+ t_0 (* (cos y) (/ (- 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 = pow(sin(y), 2.0);
	double tmp;
	if (y <= -2.9e-5) {
		tmp = (2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (t_0 + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	} else if (y <= 0.0007) {
		tmp = (0.3333333333333333 * fma(-0.0625, ((cos(x) + -1.0) * (sqrt(2.0) * pow(sin(x), 2.0))), 2.0)) / (1.0 + (1.5 + fma(cos(x), (sqrt(1.25) + -0.5), -sqrt(1.25))));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * t_1)))) / (3.0 * (t_0 + (cos(y) * ((3.0 - 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 = sin(y) ^ 2.0
	tmp = 0.0
	if (y <= -2.9e-5)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	elseif (y <= 0.0007)
		tmp = Float64(Float64(0.3333333333333333 * fma(-0.0625, Float64(Float64(cos(x) + -1.0) * Float64(sqrt(2.0) * (sin(x) ^ 2.0))), 2.0)) / Float64(1.0 + Float64(1.5 + fma(cos(x), Float64(sqrt(1.25) + -0.5), Float64(-sqrt(1.25))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * t_1)))) / Float64(3.0 * Float64(t_0 + Float64(cos(y) * Float64(Float64(3.0 - sqrt(5.0)) / 2.0)))));
	end
	return 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[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y, -2.9e-5], N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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], If[LessEqual[y, 0.0007], N[(N[(0.3333333333333333 * 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] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.5 + N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[1.25], $MachinePrecision] + -0.5), $MachinePrecision] + (-N[Sqrt[1.25], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9e-5

   1. Initial program 99.2%

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

      1. flip-- 68.5%

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

      2. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   4. Applied egg-rr 99.3%

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

      1. +-commutative 68.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 69.8%

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

   if -2.9e-5 < y < 6.99999999999999993e-4

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. +-commutative 98.6%

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

      3. fma-define 98.6%

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

      4. *-commutative 98.6%

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

      5. *-commutative 98.6%

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

      6. associate-*l* 98.6%

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

      7. sub-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

   7. Simplified 98.6%

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

      1. fma-undefine 98.6%

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

      2. fma-undefine 98.6%

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

      3. metadata-eval 98.6%

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

      4. distribute-rgt-neg-in 98.6%

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

      5. associate-+r+ 98.6%

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

   9. Applied egg-rr 98.6%

      \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(-\sqrt{1.25}\right)\right) + 1.5\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 98.6%

          \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(1.5 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(-\sqrt{1.25}\right)\right)\right)}} \]

       2. fma-define 98.6%

          \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \color{blue}{\mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, -\sqrt{1.25}\right)}\right)} \]

       3. +-commutative 98.6%

          \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \mathsf{fma}\left(\cos x, \color{blue}{-0.5 + \sqrt{1.25}}, -\sqrt{1.25}\right)\right)} \]

   11. Simplified 98.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(1.5 + \mathsf{fma}\left(\cos x, -0.5 + \sqrt{1.25}, -\sqrt{1.25}\right)\right)}} \]

   if 6.99999999999999993e-4 < y

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. +-commutative 54.9%

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

      2. associate-*r* 54.9%

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

      3. distribute-rgt-out 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around 0 63.4%

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

      1. associate-*r* 63.4%

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

      2. *-commutative 63.4%

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

   8. Simplified 63.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \mathbf{elif}\;y \leq 0.0007:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \mathsf{fma}\left(\cos x, \sqrt{1.25} + -0.5, -\sqrt{1.25}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -106 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \cos y \cdot \left(1.5 - t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -106.0) (not (<= x 1.9e-8)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (*
       3.0
       (+
        (+ 1.0 (* (cos x) (/ (+ (sqrt 5.0) -1.0) 2.0)))
        (* (cos y) (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 2.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -106.0) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-106.0d0)) .or. (.not. (x <= 1.9d-8))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + (cos(y) * (1.5d0 - t_0)))))
    else
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 * ((1.0d0 + (cos(x) * ((sqrt(5.0d0) + (-1.0d0)) / 2.0d0))) + (cos(y) * ((4.0d0 / (3.0d0 + sqrt(5.0d0))) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -106.0) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + (Math.cos(y) * (1.5 - t_0)))));
	} else {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 * ((1.0 + (Math.cos(x) * ((Math.sqrt(5.0) + -1.0) / 2.0))) + (Math.cos(y) * ((4.0 / (3.0 + Math.sqrt(5.0))) / 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -106.0) or not (x <= 1.9e-8):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + ((math.cos(x) * (t_0 - 0.5)) + (math.cos(y) * (1.5 - t_0)))))
	else:
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 * ((1.0 + (math.cos(x) * ((math.sqrt(5.0) + -1.0) / 2.0))) + (math.cos(y) * ((4.0 / (3.0 + math.sqrt(5.0))) / 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -106.0) || !(x <= 1.9e-8))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(cos(y) * Float64(1.5 - t_0))))));
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) + -1.0) / 2.0))) + Float64(cos(y) * Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -106.0) || ~((x <= 1.9e-8)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + (cos(y) * (1.5 - t_0)))));
	else
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) + -1.0) / 2.0))) + (cos(y) * ((4.0 / (3.0 + sqrt(5.0))) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -106.0], N[Not[LessEqual[x, 1.9e-8]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -106 or 1.90000000000000014e-8 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 65.1%

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

      1. associate-*r* 65.1%

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

      2. *-commutative 65.1%

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

   7. Simplified 65.1%

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

   8. Taylor expanded in y around 0 65.1%

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

   9. Taylor expanded in x around inf 65.2%

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

   if -106 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   4. Applied egg-rr 99.7%

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

      1. +-commutative 98.5%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 98.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -106 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{\frac{4}{3 + \sqrt{5}}}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := \cos y \cdot \left(1.5 - t\_0\right)\\ \mathbf{if}\;x \leq -106 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t\_0 + t\_1\right) - 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)) (t_1 (* (cos y) (- 1.5 t_0))))
   (if (or (<= x -106.0) (not (<= x 1.9e-8)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (+ 1.0 (+ (* (cos x) (- t_0 0.5)) t_1))))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))))
      (* 3.0 (+ 1.0 (- (+ t_0 t_1) 0.5)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = cos(y) * (1.5 - t_0);
	double tmp;
	if ((x <= -106.0) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + t_1)));
	} else {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + t_1) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    t_1 = cos(y) * (1.5d0 - t_0)
    if ((x <= (-106.0d0)) .or. (.not. (x <= 1.9d-8))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + ((cos(x) * (t_0 - 0.5d0)) + t_1)))
    else
        tmp = (2.0d0 + ((cos(x) - cos(y)) * (sqrt(2.0d0) * ((-0.0625d0) * (sin(y) ** 2.0d0))))) / (3.0d0 * (1.0d0 + ((t_0 + t_1) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double t_1 = Math.cos(y) * (1.5 - t_0);
	double tmp;
	if ((x <= -106.0) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + ((Math.cos(x) * (t_0 - 0.5)) + t_1)));
	} else {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (Math.sqrt(2.0) * (-0.0625 * Math.pow(Math.sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + t_1) - 0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	t_1 = math.cos(y) * (1.5 - t_0)
	tmp = 0
	if (x <= -106.0) or not (x <= 1.9e-8):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + ((math.cos(x) * (t_0 - 0.5)) + t_1)))
	else:
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (math.sqrt(2.0) * (-0.0625 * math.pow(math.sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + t_1) - 0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = Float64(cos(y) * Float64(1.5 - t_0))
	tmp = 0.0
	if ((x <= -106.0) || !(x <= 1.9e-8))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + t_1))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + t_1) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	t_1 = cos(y) * (1.5 - t_0);
	tmp = 0.0;
	if ((x <= -106.0) || ~((x <= 1.9e-8)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + t_1)));
	else
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * (sin(y) ^ 2.0))))) / (3.0 * (1.0 + ((t_0 + t_1) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -106.0], N[Not[LessEqual[x, 1.9e-8]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + t$95$1), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -106 or 1.90000000000000014e-8 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 65.1%

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

      1. associate-*r* 65.1%

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

      2. *-commutative 65.1%

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

   7. Simplified 65.1%

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

   8. Taylor expanded in y around 0 65.1%

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

   9. Taylor expanded in x around inf 65.2%

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

   if -106 < x < 1.90000000000000014e-8

   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%

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

      2. distribute-rgt-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-rgt-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 98.6%

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

   6. Taylor expanded in x around 0 98.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -106 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ t_1 := {\sin x}^{2}\\ t_2 := \cos x + -1\\ \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, t\_2 \cdot \left(\sqrt{2} \cdot t\_1\right), 2\right)}{1 + \left(1.5 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) - \sqrt{1.25}\right)\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(t\_0 + \cos y \cdot \left(1.5 - t\_0\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t\_1 \cdot \left(\sqrt{2} \cdot t\_2\right)\right)}{1 + \left(\cos x \cdot \left(t\_0 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5))
        (t_1 (pow (sin x) 2.0))
        (t_2 (+ (cos x) -1.0)))
   (if (<= x -0.00072)
     (/
      (* 0.3333333333333333 (fma -0.0625 (* t_2 (* (sqrt 2.0) t_1)) 2.0))
      (+ 1.0 (+ 1.5 (- (* (cos x) (+ (sqrt 1.25) -0.5)) (sqrt 1.25)))))
     (if (<= x 1.9e-8)
       (/
        (+
         2.0
         (* (- (cos x) (cos y)) (* (sqrt 2.0) (* -0.0625 (pow (sin y) 2.0)))))
        (* 3.0 (+ 1.0 (- (+ t_0 (* (cos y) (- 1.5 t_0))) 0.5))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* t_1 (* (sqrt 2.0) t_2))))
         (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (- 3.0 (sqrt 5.0)) 0.5)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double t_1 = pow(sin(x), 2.0);
	double t_2 = cos(x) + -1.0;
	double tmp;
	if (x <= -0.00072) {
		tmp = (0.3333333333333333 * fma(-0.0625, (t_2 * (sqrt(2.0) * t_1)), 2.0)) / (1.0 + (1.5 + ((cos(x) * (sqrt(1.25) + -0.5)) - sqrt(1.25))));
	} else if (x <= 1.9e-8) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (sqrt(2.0) * (-0.0625 * pow(sin(y), 2.0))))) / (3.0 * (1.0 + ((t_0 + (cos(y) * (1.5 - t_0))) - 0.5)));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_1 * (sqrt(2.0) * t_2)))) / (1.0 + ((cos(x) * (t_0 - 0.5)) + ((3.0 - sqrt(5.0)) * 0.5))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(cos(x) + -1.0)
	tmp = 0.0
	if (x <= -0.00072)
		tmp = Float64(Float64(0.3333333333333333 * fma(-0.0625, Float64(t_2 * Float64(sqrt(2.0) * t_1)), 2.0)) / Float64(1.0 + Float64(1.5 + Float64(Float64(cos(x) * Float64(sqrt(1.25) + -0.5)) - sqrt(1.25)))));
	elseif (x <= 1.9e-8)
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(sqrt(2.0) * Float64(-0.0625 * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(1.0 + Float64(Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0))) - 0.5))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_1 * Float64(sqrt(2.0) * t_2)))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + Float64(Float64(3.0 - sqrt(5.0)) * 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.00072], N[(N[(0.3333333333333333 * N[(-0.0625 * N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[1.25], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-8], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(1.0 + N[(N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.20000000000000045e-4

   1. Initial program 98.9%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around 0 62.1%

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

      1. associate-*r/ 62.3%

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

      2. +-commutative 62.3%

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

      3. fma-define 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. associate-*l* 62.3%

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

      7. sub-neg 62.3%

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

      8. metadata-eval 62.3%

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

      9. +-commutative 62.3%

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

   7. Simplified 62.3%

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

      1. fma-undefine 62.3%

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

      2. fma-undefine 62.3%

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

      3. metadata-eval 62.3%

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

      4. distribute-rgt-neg-in 62.3%

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

      5. associate-+r+ 62.2%

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

   9. Applied egg-rr 62.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(-\sqrt{1.25}\right)\right) + 1.5\right)}} \]

   if -7.20000000000000045e-4 < x < 1.90000000000000014e-8

   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%

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

      2. distribute-rgt-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-rgt-in 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1.90000000000000014e-8 < x

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 65.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) - \sqrt{1.25}\right)\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\sqrt{2} \cdot \left(-0.0625 \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(1 + \left(\left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right) - 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := \cos x + -1\\ \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, t\_1 \cdot \left(\sqrt{2} \cdot t\_0\right), 2\right)}{1 + \left(1.5 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) - \sqrt{1.25}\right)\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot t\_1\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)) (t_1 (+ (cos x) -1.0)))
   (if (<= x -0.00072)
     (/
      (* 0.3333333333333333 (fma -0.0625 (* t_1 (* (sqrt 2.0) t_0)) 2.0))
      (+ 1.0 (+ 1.5 (- (* (cos x) (+ (sqrt 1.25) -0.5)) (sqrt 1.25)))))
     (if (<= x 1.9e-8)
       (*
        0.3333333333333333
        (/
         (+
          2.0
          (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
         (+
          0.5
          (* 0.5 (+ (sqrt 5.0) (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))))))))
       (*
        0.3333333333333333
        (/
         (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) t_1))))
         (+
          1.0
          (+
           (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5))
           (* (- 3.0 (sqrt 5.0)) 0.5)))))))))

C

double code(double x, double y) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = cos(x) + -1.0;
	double tmp;
	if (x <= -0.00072) {
		tmp = (0.3333333333333333 * fma(-0.0625, (t_1 * (sqrt(2.0) * t_0)), 2.0)) / (1.0 + (1.5 + ((cos(x) * (sqrt(1.25) + -0.5)) - sqrt(1.25))));
	} else if (x <= 1.9e-8) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (0.5 * (sqrt(5.0) + (cos(y) * (4.0 / (3.0 + sqrt(5.0))))))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * t_1)))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - sqrt(5.0)) * 0.5))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = sin(x) ^ 2.0
	t_1 = Float64(cos(x) + -1.0)
	tmp = 0.0
	if (x <= -0.00072)
		tmp = Float64(Float64(0.3333333333333333 * fma(-0.0625, Float64(t_1 * Float64(sqrt(2.0) * t_0)), 2.0)) / Float64(1.0 + Float64(1.5 + Float64(Float64(cos(x) * Float64(sqrt(1.25) + -0.5)) - sqrt(1.25)))));
	elseif (x <= 1.9e-8)
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(0.5 + Float64(0.5 * Float64(sqrt(5.0) + Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * t_1)))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)) + Float64(Float64(3.0 - sqrt(5.0)) * 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, -0.00072], N[(N[(0.3333333333333333 * N[(-0.0625 * N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.5 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[1.25], $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[1.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-8], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.20000000000000045e-4

   1. Initial program 98.9%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in y around 0 62.1%

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

      1. associate-*r/ 62.3%

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

      2. +-commutative 62.3%

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

      3. fma-define 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. associate-*l* 62.3%

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

      7. sub-neg 62.3%

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

      8. metadata-eval 62.3%

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

      9. +-commutative 62.3%

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

   7. Simplified 62.3%

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

      1. fma-undefine 62.3%

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

      2. fma-undefine 62.3%

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

      3. metadata-eval 62.3%

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

      4. distribute-rgt-neg-in 62.3%

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

      5. associate-+r+ 62.2%

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

   9. Applied egg-rr 62.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \color{blue}{\left(\left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) + \left(-\sqrt{1.25}\right)\right) + 1.5\right)}} \]

   if -7.20000000000000045e-4 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-lft-out 99.1%

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

   7. Simplified 99.1%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   9. Applied egg-rr 99.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 99.1%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   11. Simplified 99.1%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}}\right)} \]

   if 1.90000000000000014e-8 < x

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around 0 65.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00072:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \mathsf{fma}\left(-0.0625, \left(\cos x + -1\right) \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right), 2\right)}{1 + \left(1.5 + \left(\cos x \cdot \left(\sqrt{1.25} + -0.5\right) - \sqrt{1.25}\right)\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00072 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.00072) (not (<= x 1.9e-8)))
   (*
    0.3333333333333333
    (/
     (+ 2.0 (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
     (+
      1.0
      (+ (* (cos x) (- (* (sqrt 5.0) 0.5) 0.5)) (* (- 3.0 (sqrt 5.0)) 0.5)))))
   (*
    0.3333333333333333
    (/
     (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
     (+ 0.5 (* 0.5 (+ (sqrt 5.0) (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.00072) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - sqrt(5.0)) * 0.5))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (0.5 * (sqrt(5.0) + (cos(y) * (4.0 / (3.0 + sqrt(5.0))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-0.00072d0)) .or. (.not. (x <= 1.9d-8))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (1.0d0 + ((cos(x) * ((sqrt(5.0d0) * 0.5d0) - 0.5d0)) + ((3.0d0 - sqrt(5.0d0)) * 0.5d0))))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (0.5d0 + (0.5d0 * (sqrt(5.0d0) + (cos(y) * (4.0d0 / (3.0d0 + sqrt(5.0d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.00072) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (1.0 + ((Math.cos(x) * ((Math.sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - Math.sqrt(5.0)) * 0.5))));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (0.5 + (0.5 * (Math.sqrt(5.0) + (Math.cos(y) * (4.0 / (3.0 + Math.sqrt(5.0))))))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.00072) or not (x <= 1.9e-8):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (1.0 + ((math.cos(x) * ((math.sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - math.sqrt(5.0)) * 0.5))))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (0.5 + (0.5 * (math.sqrt(5.0) + (math.cos(y) * (4.0 / (3.0 + math.sqrt(5.0))))))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.00072) || !(x <= 1.9e-8))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(1.0 + Float64(Float64(cos(x) * Float64(Float64(sqrt(5.0) * 0.5) - 0.5)) + Float64(Float64(3.0 - sqrt(5.0)) * 0.5)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(0.5 + Float64(0.5 * Float64(sqrt(5.0) + Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.00072) || ~((x <= 1.9e-8)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (1.0 + ((cos(x) * ((sqrt(5.0) * 0.5) - 0.5)) + ((3.0 - sqrt(5.0)) * 0.5))));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (0.5 * (sqrt(5.0) + (cos(y) * (4.0 / (3.0 + sqrt(5.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.00072], N[Not[LessEqual[x, 1.9e-8]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000045e-4 or 1.90000000000000014e-8 < x

   1. Initial program 98.9%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in y around 0 64.0%

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

   if -7.20000000000000045e-4 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-lft-out 99.1%

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

   7. Simplified 99.1%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   9. Applied egg-rr 99.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 99.1%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   11. Simplified 99.1%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00072 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{1 + \left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + \left(3 - \sqrt{5}\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.00072 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(t\_0 - 0.5\right) + 2.5\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(1 - {\cos y}^{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.00072) (not (<= x 1.9e-8)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (- (+ (* (cos x) (- t_0 0.5)) 2.5) t_0)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (*
         -0.0625
         (* (* (sqrt 2.0) (- 1.0 (cos y))) (- 1.0 (pow (cos y) 2.0)))))
       (+ 0.5 (* 0.5 (+ (sqrt 5.0) (* (cos y) (- 3.0 (sqrt 5.0)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00072) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (1.0 - cos(y))) * (1.0 - pow(cos(y), 2.0))))) / (0.5 + (0.5 * (sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-0.00072d0)) .or. (.not. (x <= 1.9d-8))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (((cos(x) * (t_0 - 0.5d0)) + 2.5d0) - t_0))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sqrt(2.0d0) * (1.0d0 - cos(y))) * (1.0d0 - (cos(y) ** 2.0d0))))) / (0.5d0 + (0.5d0 * (sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00072) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (((Math.cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((Math.sqrt(2.0) * (1.0 - Math.cos(y))) * (1.0 - Math.pow(Math.cos(y), 2.0))))) / (0.5 + (0.5 * (Math.sqrt(5.0) + (Math.cos(y) * (3.0 - Math.sqrt(5.0)))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -0.00072) or not (x <= 1.9e-8):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (((math.cos(x) * (t_0 - 0.5)) + 2.5) - t_0))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((math.sqrt(2.0) * (1.0 - math.cos(y))) * (1.0 - math.pow(math.cos(y), 2.0))))) / (0.5 + (0.5 * (math.sqrt(5.0) + (math.cos(y) * (3.0 - math.sqrt(5.0)))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -0.00072) || !(x <= 1.9e-8))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + 2.5) - t_0)));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64(Float64(sqrt(2.0) * Float64(1.0 - cos(y))) * Float64(1.0 - (cos(y) ^ 2.0))))) / Float64(0.5 + Float64(0.5 * Float64(sqrt(5.0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -0.00072) || ~((x <= 1.9e-8)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sqrt(2.0) * (1.0 - cos(y))) * (1.0 - (cos(y) ^ 2.0))))) / (0.5 + (0.5 * (sqrt(5.0) + (cos(y) * (3.0 - sqrt(5.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -0.00072], N[Not[LessEqual[x, 1.9e-8]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + 2.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[Cos[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000045e-4 or 1.90000000000000014e-8 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in y around 0 64.8%

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

   9. Taylor expanded in y around 0 63.9%

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

   if -7.20000000000000045e-4 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-lft-out 99.1%

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

   7. Simplified 99.1%

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

      1. unpow2 99.1%

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

      2. 1-sub-cos 99.1%

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

      3. sub-neg 99.1%

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

      4. pow2 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. sub-neg 99.1%

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

   11. Simplified 99.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00072 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 2.5\right) - \sqrt{5} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\sqrt{2} \cdot \left(1 - \cos y\right)\right) \cdot \left(1 - {\cos y}^{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 78.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \mathbf{if}\;x \leq -0.00074 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(t\_0 - 0.5\right) + 2.5\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (if (or (<= x -0.00074) (not (<= x 1.9e-8)))
     (*
      0.3333333333333333
      (/
       (+
        2.0
        (* -0.0625 (* (pow (sin x) 2.0) (* (sqrt 2.0) (+ (cos x) -1.0)))))
       (- (+ (* (cos x) (- t_0 0.5)) 2.5) t_0)))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+
        0.5
        (* 0.5 (+ (sqrt 5.0) (* (cos y) (/ 4.0 (+ 3.0 (sqrt 5.0))))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00074) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(x), 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (0.5 * (sqrt(5.0) + (cos(y) * (4.0 / (3.0 + sqrt(5.0))))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(5.0d0) * 0.5d0
    if ((x <= (-0.00074d0)) .or. (.not. (x <= 1.9d-8))) then
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(x) ** 2.0d0) * (sqrt(2.0d0) * (cos(x) + (-1.0d0)))))) / (((cos(x) * (t_0 - 0.5d0)) + 2.5d0) - t_0))
    else
        tmp = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (0.5d0 + (0.5d0 * (sqrt(5.0d0) + (cos(y) * (4.0d0 / (3.0d0 + sqrt(5.0d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	double tmp;
	if ((x <= -0.00074) || !(x <= 1.9e-8)) {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(x), 2.0) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0))))) / (((Math.cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	} else {
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (0.5 + (0.5 * (Math.sqrt(5.0) + (Math.cos(y) * (4.0 / (3.0 + Math.sqrt(5.0))))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	tmp = 0
	if (x <= -0.00074) or not (x <= 1.9e-8):
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(x), 2.0) * (math.sqrt(2.0) * (math.cos(x) + -1.0))))) / (((math.cos(x) * (t_0 - 0.5)) + 2.5) - t_0))
	else:
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (0.5 + (0.5 * (math.sqrt(5.0) + (math.cos(y) * (4.0 / (3.0 + math.sqrt(5.0))))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	tmp = 0.0
	if ((x <= -0.00074) || !(x <= 1.9e-8))
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0))))) / Float64(Float64(Float64(cos(x) * Float64(t_0 - 0.5)) + 2.5) - t_0)));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(2.0 + Float64(-0.0625 * Float64((sin(y) ^ 2.0) * Float64(sqrt(2.0) * Float64(1.0 - cos(y)))))) / Float64(0.5 + Float64(0.5 * Float64(sqrt(5.0) + Float64(cos(y) * Float64(4.0 / Float64(3.0 + sqrt(5.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.0;
	if ((x <= -0.00074) || ~((x <= 1.9e-8)))
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(x) ^ 2.0) * (sqrt(2.0) * (cos(x) + -1.0))))) / (((cos(x) * (t_0 - 0.5)) + 2.5) - t_0));
	else
		tmp = 0.3333333333333333 * ((2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (0.5 + (0.5 * (sqrt(5.0) + (cos(y) * (4.0 / (3.0 + sqrt(5.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[x, -0.00074], N[Not[LessEqual[x, 1.9e-8]], $MachinePrecision]], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision] + 2.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(2.0 + N[(-0.0625 * N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[(N[Sqrt[5.0], $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.3999999999999999e-4 or 1.90000000000000014e-8 < x

   1. Initial program 98.9%

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

      1. associate-*l* 98.9%

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

      2. distribute-rgt-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-rgt-in 98.9%

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

      5. associate-+l+ 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in y around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in y around 0 64.8%

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

   9. Taylor expanded in y around 0 63.9%

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

   if -7.3999999999999999e-4 < x < 1.90000000000000014e-8

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-lft-out 99.1%

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

   7. Simplified 99.1%

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

      1. flip-- 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{9} - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      3. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{0.5}} \cdot \sqrt{5}}{3 + \sqrt{5}}\right)} \]

      4. pow1/2 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{0.5} \cdot \color{blue}{{5}^{0.5}}}{3 + \sqrt{5}}\right)} \]

      5. pow-prod-up 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{{5}^{\left(0.5 + 0.5\right)}}}{3 + \sqrt{5}}\right)} \]

      6. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - {5}^{\color{blue}{1}}}{3 + \sqrt{5}}\right)} \]

      7. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{9 - \color{blue}{5}}{3 + \sqrt{5}}\right)} \]

      8. metadata-eval 99.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{\color{blue}{4}}{3 + \sqrt{5}}\right)} \]

   9. Applied egg-rr 99.1%

      \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \color{blue}{\frac{4}{3 + \sqrt{5}}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 99.1%

          \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{\color{blue}{\sqrt{5} + 3}}\right)} \]

   11. Simplified 99.1%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \color{blue}{\frac{4}{\sqrt{5} + 3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00074 \lor \neg \left(x \leq 1.9 \cdot 10^{-8}\right):\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)\right)}{\left(\cos x \cdot \left(\sqrt{5} \cdot 0.5 - 0.5\right) + 2.5\right) - \sqrt{5} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \frac{4}{3 + \sqrt{5}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 59.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sqrt(2.0d0) * (1.0d0 - cos(y))) * (1.0d0 - (cos(y) ** 2.0d0))))) / (0.5d0 + (0.5d0 * (sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 61.4%

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

   1. *-commutative 61.4%

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

   2. distribute-lft-out 61.4%

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

7. Simplified 61.4%

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

   1. unpow2 61.4%

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

   2. 1-sub-cos 61.4%

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

   3. sub-neg 61.4%

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

   4. pow2 61.4%

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

9. Applied egg-rr 61.4%

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

    1. sub-neg 61.4%

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

11. Simplified 61.4%

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

12. Final simplification 61.4%

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

Alternative 22: 59.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0 * ((2.0d0 + ((-0.0625d0) * ((sqrt(2.0d0) * (1.0d0 - cos(y))) * (0.5d0 - (cos((2.0d0 * y)) / 2.0d0))))) / (0.5d0 + (0.5d0 * (sqrt(5.0d0) + (cos(y) * (3.0d0 - sqrt(5.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 61.4%

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

   1. *-commutative 61.4%

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

   2. distribute-lft-out 61.4%

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

7. Simplified 61.4%

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

   1. unpow2 61.4%

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

   2. sin-mult 61.4%

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

9. Applied egg-rr 61.4%

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

    1. div-sub 61.4%

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

    2. +-inverses 61.4%

       \[\leadsto 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(y + y\right)}{2}\right) \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{0.5 + 0.5 \cdot \left(\sqrt{5} + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \]

    3. cos-0 61.4%

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

    4. metadata-eval 61.4%

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

    5. count-2 61.4%

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

    6. *-commutative 61.4%

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

11. Simplified 61.4%

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

12. Final simplification 61.4%

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

Alternative 23: 41.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} \cdot 0.5\\ \frac{0.6666666666666666}{0.5 + \left(t\_0 + \cos y \cdot \left(1.5 - t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt 5.0) 0.5)))
   (/ 0.6666666666666666 (+ 0.5 (+ t_0 (* (cos y) (- 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) * 0.5;
	return 0.6666666666666666 / (0.5 + (t_0 + (cos(y) * (1.5 - t_0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) * 0.5d0
    code = 0.6666666666666666d0 / (0.5d0 + (t_0 + (cos(y) * (1.5d0 - t_0))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) * 0.5;
	return 0.6666666666666666 / (0.5 + (t_0 + (Math.cos(y) * (1.5 - t_0))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) * 0.5
	return 0.6666666666666666 / (0.5 + (t_0 + (math.cos(y) * (1.5 - t_0))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) * 0.5)
	return Float64(0.6666666666666666 / Float64(0.5 + Float64(t_0 + Float64(cos(y) * Float64(1.5 - t_0)))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) * 0.5;
	tmp = 0.6666666666666666 / (0.5 + (t_0 + (cos(y) * (1.5 - t_0))));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(0.6666666666666666 / N[(0.5 + N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

   5. associate-+l+ 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in y around 0 62.5%

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

   1. associate-*r* 62.5%

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

   2. *-commutative 62.5%

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

7. Simplified 62.5%

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

8. Taylor expanded in y around 0 62.5%

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

9. Taylor expanded in x around 0 41.6%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{0.5 + \left(0.5 \cdot \sqrt{5} + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right)}} \]

10. Final simplification 41.6%

    \[\leadsto \frac{0.6666666666666666}{0.5 + \left(\sqrt{5} \cdot 0.5 + \cos y \cdot \left(1.5 - \sqrt{5} \cdot 0.5\right)\right)} \]
11. Add Preprocessing

Alternative 24: 40.0% accurate, 1139.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 61.4%

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

   1. *-commutative 61.4%

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

   2. distribute-lft-out 61.4%

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

7. Simplified 61.4%

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

8. Taylor expanded in y around 0 29.7%

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

9. Taylor expanded in y around 0 29.4%

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

10. Taylor expanded in y around 0 39.6%

    \[\leadsto \color{blue}{0.3333333333333333} \]

11. Final simplification 39.6%

    \[\leadsto 0.3333333333333333 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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))))))