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

Percentage Accurate: 99.3% → 99.4%

Time: 1.0min

Alternatives: 22

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \sqrt{5} \cdot 0.6666666666666666\\ t_1 := 0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \frac{\left(\cos y \cdot 4\right) \cdot t_1 + \cos x \cdot t_0}{t_1 \cdot t_0}} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* (sqrt 5.0) 0.6666666666666666)))
        (t_1 (* 0.16666666666666666 (+ (sqrt 5.0) 1.0))))
   (/
    (fma
     (sqrt 2.0)
     (*
      (+ (sin x) (* -0.0625 (sin y)))
      (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y))))
     2.0)
    (+ 3.0 (/ (+ (* (* (cos y) 4.0) t_1) (* (cos x) t_0)) (* t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + (sqrt(5.0) * 0.6666666666666666);
	double t_1 = 0.16666666666666666 * (sqrt(5.0) + 1.0);
	return fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y)))), 2.0) / (3.0 + ((((cos(y) * 4.0) * t_1) + (cos(x) * t_0)) / (t_1 * t_0)));
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + Float64(sqrt(5.0) * 0.6666666666666666))
	t_1 = Float64(0.16666666666666666 * Float64(sqrt(5.0) + 1.0))
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(Float64(Float64(cos(y) * 4.0) * t_1) + Float64(cos(x) * t_0)) / Float64(t_1 * t_0))))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.16666666666666666 * N[(N[Sqrt[5.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[(N[(N[(N[Cos[y], $MachinePrecision] * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $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%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]
4. Step-by-step derivation

   1. flip-- 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   2. sub-neg 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{3 \cdot 3 + \left(-\sqrt{5} \cdot \sqrt{5}\right)}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   3. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. rem-square-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   7. +-commutative 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

5. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
6. Step-by-step derivation

   1. fma-udef 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   2. associate-/l/ 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \color{blue}{\frac{4}{0.6666666666666666 \cdot \left(\sqrt{5} + 3\right)}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   3. associate-*r/ 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\color{blue}{\frac{\cos y \cdot 4}{0.6666666666666666 \cdot \left(\sqrt{5} + 3\right)}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. associate-/l* 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y \cdot 4}{0.6666666666666666 \cdot \left(\sqrt{5} + 3\right)} + \color{blue}{\frac{\cos x}{\frac{0.6666666666666666}{\sqrt{5} + -1}}}\right)} \]

   5. frac-add 99.2%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\frac{\left(\cos y \cdot 4\right) \cdot \frac{0.6666666666666666}{\sqrt{5} + -1} + \left(0.6666666666666666 \cdot \left(\sqrt{5} + 3\right)\right) \cdot \cos x}{\left(0.6666666666666666 \cdot \left(\sqrt{5} + 3\right)\right) \cdot \frac{0.6666666666666666}{\sqrt{5} + -1}}}} \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\frac{\left(\cos y \cdot 4\right) \cdot \left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right) + \left(2 + 0.6666666666666666 \cdot \sqrt{5}\right) \cdot \cos x}{\left(2 + 0.6666666666666666 \cdot \sqrt{5}\right) \cdot \left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right)}}} \]

8. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \frac{\left(\cos y \cdot 4\right) \cdot \left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right) + \cos x \cdot \left(2 + \sqrt{5} \cdot 0.6666666666666666\right)}{\left(0.16666666666666666 \cdot \left(\sqrt{5} + 1\right)\right) \cdot \left(2 + \sqrt{5} \cdot 0.6666666666666666\right)}} \]

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\sqrt{5} + -3}} + \left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot 1.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin x) (* -0.0625 (sin y)))
    (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (+
    (/ (cos y) (/ -0.6666666666666666 (+ (sqrt 5.0) -3.0)))
    (* (+ (sqrt 5.0) -1.0) (* (cos x) 1.5))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y)))), 2.0) / (3.0 + ((cos(y) / (-0.6666666666666666 / (sqrt(5.0) + -3.0))) + ((sqrt(5.0) + -1.0) * (cos(x) * 1.5))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(Float64(cos(y) / Float64(-0.6666666666666666 / Float64(sqrt(5.0) + -3.0))) + Float64(Float64(sqrt(5.0) + -1.0) * Float64(cos(x) * 1.5)))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $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[(-0.6666666666666666 / N[(N[Sqrt[5.0], $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Cos[x], $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%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]
4. Step-by-step derivation

   1. flip-- 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   2. sub-neg 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{3 \cdot 3 + \left(-\sqrt{5} \cdot \sqrt{5}\right)}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   3. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. rem-square-sqrt 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   7. +-commutative 99.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

5. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]
6. Step-by-step derivation

   1. fma-udef 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\cos y \cdot \frac{\frac{4}{\sqrt{5} + 3}}{0.6666666666666666} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   2. clear-num 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\cos y \cdot \color{blue}{\frac{1}{\frac{0.6666666666666666}{\frac{4}{\sqrt{5} + 3}}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   3. un-div-inv 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\color{blue}{\frac{\cos y}{\frac{0.6666666666666666}{\frac{4}{\sqrt{5} + 3}}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   4. frac-2neg 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\color{blue}{\frac{-0.6666666666666666}{-\frac{4}{\sqrt{5} + 3}}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{\color{blue}{-0.6666666666666666}}{-\frac{4}{\sqrt{5} + 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. distribute-neg-frac 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\color{blue}{\frac{-4}{\sqrt{5} + 3}}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   7. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\frac{\color{blue}{-4}}{\sqrt{5} + 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   8. metadata-eval 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\frac{\color{blue}{5 + -9}}{\sqrt{5} + 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   9. rem-square-sqrt 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\frac{\color{blue}{\sqrt{5} \cdot \sqrt{5}} + -9}{\sqrt{5} + 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   10. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} + \color{blue}{\left(-9\right)}}{\sqrt{5} + 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   11. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\frac{\sqrt{5} \cdot \sqrt{5} + \left(-\color{blue}{3 \cdot 3}\right)}{\sqrt{5} + 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   12. sub-neg 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\frac{\color{blue}{\sqrt{5} \cdot \sqrt{5} - 3 \cdot 3}}{\sqrt{5} + 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   13. flip-- 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\color{blue}{\sqrt{5} - 3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   14. sub-neg 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\color{blue}{\sqrt{5} + \left(-3\right)}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   15. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\sqrt{5} + \color{blue}{-3}}} + \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   16. div-inv 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\sqrt{5} + -3}} + \color{blue}{\left(\cos x \cdot \left(\sqrt{5} + -1\right)\right) \cdot \frac{1}{0.6666666666666666}}\right)} \]

   17. *-commutative 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\sqrt{5} + -3}} + \color{blue}{\left(\left(\sqrt{5} + -1\right) \cdot \cos x\right)} \cdot \frac{1}{0.6666666666666666}\right)} \]

   18. metadata-eval 99.4%

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\sqrt{5} + -3}} + \left(\left(\sqrt{5} + -1\right) \cdot \cos x\right) \cdot \color{blue}{1.5}\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \color{blue}{\left(\frac{\cos y}{\frac{-0.6666666666666666}{\sqrt{5} + -3}} + \left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot 1.5\right)\right)}} \]

8. Final simplification 99.4%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \left(\frac{\cos y}{\frac{-0.6666666666666666}{\sqrt{5} + -3}} + \left(\sqrt{5} + -1\right) \cdot \left(\cos x \cdot 1.5\right)\right)} \]

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\cos x \cdot \left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (fma
   (sqrt 2.0)
   (*
    (+ (sin x) (* -0.0625 (sin y)))
    (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y))))
   2.0)
  (+
   3.0
   (*
    1.5
    (+ (* (cos x) (+ (sqrt 5.0) -1.0)) (* (cos y) (- 3.0 (sqrt 5.0))))))))

C

double code(double x, double y) {
	return fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y)))), 2.0) / (3.0 + (1.5 * ((cos(x) * (sqrt(5.0) + -1.0)) + (cos(y) * (3.0 - sqrt(5.0))))));
}

Julia

function code(x, y)
	return Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))))
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

4. Taylor expanded in y around inf 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (/
    (+
     2.0
     (*
      (* (- (sin y) (/ (sin x) 16.0)) (- (cos x) (cos y)))
      (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))))
    (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	return (2.0 + (((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) / 2.0d0
    code = (2.0d0 + (((sin(y) - (sin(x) / 16.0d0)) * (cos(x) - cos(y))) * (sqrt(2.0d0) * (sin(x) - (sin(y) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	return (2.0 + (((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.cos(x) - Math.cos(y))) * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	return (2.0 + (((math.sin(y) - (math.sin(x) / 16.0)) * (math.cos(x) - math.cos(y))) * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	return Float64(Float64(2.0 + Float64(Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(cos(x) - cos(y))) * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = (2.0 + (((sin(y) - (sin(x) / 16.0)) * (cos(x) - cos(y))) * (sqrt(2.0) * (sin(x) - (sin(y) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/
  (+
   2.0
   (*
    (- (cos x) (cos y))
    (*
     (- (sin y) (/ (sin x) 16.0))
     (* (sqrt 2.0) (- (sin x) (/ (sin y) 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)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 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)) * ((sin(y) - (sin(x) / 16.0d0)) * (sqrt(2.0d0) * (sin(x) - (sin(y) / 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.sin(y) - (Math.sin(x) / 16.0)) * (Math.sqrt(2.0) * (Math.sin(x) - (Math.sin(y) / 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.sin(y) - (math.sin(x) / 16.0)) * (math.sqrt(2.0) * (math.sin(x) - (math.sin(y) / 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(sin(y) - Float64(sin(x) / 16.0)) * Float64(sqrt(2.0) * Float64(sin(x) - Float64(sin(y) / 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)) * ((sin(y) - (sin(x) / 16.0)) * (sqrt(2.0) * (sin(x) - (sin(y) / 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[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \frac{2 + \left(\sqrt{2} \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (/
    (+
     2.0
     (*
      (* (sqrt 2.0) (- (cos x) (cos y)))
      (* (- (sin y) (/ (sin x) 16.0)) (- (sin x) (/ (sin y) 16.0)))))
    (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	return (2.0 + ((sqrt(2.0) * (cos(x) - cos(y))) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt(5.0d0) / 2.0d0
    code = (2.0d0 + ((sqrt(2.0d0) * (cos(x) - cos(y))) * ((sin(y) - (sin(x) / 16.0d0)) * (sin(x) - (sin(y) / 16.0d0))))) / (3.0d0 * ((1.0d0 + (cos(x) * (t_0 - 0.5d0))) + (cos(y) * (1.5d0 - t_0))))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	return (2.0 + ((Math.sqrt(2.0) * (Math.cos(x) - Math.cos(y))) * ((Math.sin(y) - (Math.sin(x) / 16.0)) * (Math.sin(x) - (Math.sin(y) / 16.0))))) / (3.0 * ((1.0 + (Math.cos(x) * (t_0 - 0.5))) + (Math.cos(y) * (1.5 - t_0))));
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	return (2.0 + ((math.sqrt(2.0) * (math.cos(x) - math.cos(y))) * ((math.sin(y) - (math.sin(x) / 16.0)) * (math.sin(x) - (math.sin(y) / 16.0))))) / (3.0 * ((1.0 + (math.cos(x) * (t_0 - 0.5))) + (math.cos(y) * (1.5 - t_0))))

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	return Float64(Float64(2.0 + Float64(Float64(sqrt(2.0) * Float64(cos(x) - cos(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(sin(x) - Float64(sin(y) / 16.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	tmp = (2.0 + ((sqrt(2.0) * (cos(x) - cos(y))) * ((sin(y) - (sin(x) / 16.0)) * (sin(x) - (sin(y) / 16.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

   1. add-exp-log_binary64 65.7%

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

5. Applied rewrite-once 65.7%

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

   1. rem-exp-log 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

7. Simplified 99.3%

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

   1. add-exp-log_binary64 65.7%

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

9. Applied rewrite-once 65.7%

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

    1. rem-exp-log 99.3%

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

    2. associate-*r* 99.3%

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 7: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0265 \lor \neg \left(x \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{t_1}\\ \end{array} \end{array} \]

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 -0.0265) (not (<= x 9.2e-5)))
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* t_0 (* (sqrt 2.0) (sin x))))) t_1)
     (/
      (+
       2.0
       (*
        (* t_0 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))))
        (- (+ 1.0 (* -0.5 (* x x))) (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 <= -0.0265) || !(x <= 9.2e-5)) {
		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) * (sin(x) - (sin(y) / 16.0)))) * ((1.0 + (-0.5 * (x * x))) - 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 <= (-0.0265d0)) .or. (.not. (x <= 9.2d-5))) 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) * (sin(x) - (sin(y) / 16.0d0)))) * ((1.0d0 + ((-0.5d0) * (x * x))) - 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 <= -0.0265) || !(x <= 9.2e-5)) {
		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) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * ((1.0 + (-0.5 * (x * x))) - 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 <= -0.0265) or not (x <= 9.2e-5):
		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) * (math.sin(x) - (math.sin(y) / 16.0)))) * ((1.0 + (-0.5 * (x * x))) - 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 <= -0.0265) || !(x <= 9.2e-5))
		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(sin(x) - Float64(sin(y) / 16.0)))) * Float64(Float64(1.0 + Float64(-0.5 * Float64(x * x))) - 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 <= -0.0265) || ~((x <= 9.2e-5)))
		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) * (sin(x) - (sin(y) / 16.0)))) * ((1.0 + (-0.5 * (x * x))) - 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, -0.0265], N[Not[LessEqual[x, 9.2e-5]], $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[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0264999999999999993 or 9.20000000000000001e-5 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 65.1%

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

   if -0.0264999999999999993 < x < 9.20000000000000001e-5

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.5%

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

   4. Simplified 99.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0265 \lor \neg \left(x \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 8: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin y - \frac{\sin x}{16}\\ t_1 := 3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\\ \mathbf{if}\;x \leq -0.0055 \lor \neg \left(x \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(t_0 \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{t_1}\\ \end{array} \end{array} \]

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 -0.0055) (not (<= x 9.2e-5)))
     (/ (+ 2.0 (* (- (cos x) (cos y)) (* t_0 (* (sqrt 2.0) (sin x))))) t_1)
     (/
      (+
       2.0
       (* (* t_0 (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0)))) (- 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 <= -0.0055) || !(x <= 9.2e-5)) {
		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) * (sin(x) - (sin(y) / 16.0)))) * (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 <= (-0.0055d0)) .or. (.not. (x <= 9.2d-5))) 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) * (sin(x) - (sin(y) / 16.0d0)))) * (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 <= -0.0055) || !(x <= 9.2e-5)) {
		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) * (Math.sin(x) - (Math.sin(y) / 16.0)))) * (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 <= -0.0055) or not (x <= 9.2e-5):
		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) * (math.sin(x) - (math.sin(y) / 16.0)))) * (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 <= -0.0055) || !(x <= 9.2e-5))
		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(sin(x) - Float64(sin(y) / 16.0)))) * 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 <= -0.0055) || ~((x <= 9.2e-5)))
		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) * (sin(x) - (sin(y) / 16.0)))) * (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, -0.0055], N[Not[LessEqual[x, 9.2e-5]], $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[(N[Sin[x], $MachinePrecision] - N[(N[Sin[y], $MachinePrecision] / 16.0), $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 < -0.0054999999999999997 or 9.20000000000000001e-5 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 65.1%

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

   if -0.0054999999999999997 < x < 9.20000000000000001e-5

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0055 \lor \neg \left(x \leq 9.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(1 - \cos y\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \end{array} \]

Alternative 9: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-5} \lor \neg \left(x \leq 5.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_2}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(t_1 \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot t_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -9.6e-5) (not (<= x 5.9e-5)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* t_1 (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_2 2.0)))))
     (/
      (fma
       (sqrt 2.0)
       (*
        (+ (sin x) (* -0.0625 (sin y)))
        (* t_1 (- (+ 1.0 (* -0.5 (* x x))) (cos y))))
       2.0)
      (+ 3.0 (* 1.5 (+ t_0 (* (cos y) t_2))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -9.6e-5) || !(x <= 5.9e-5)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_1 * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_2 / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * (t_1 * ((1.0 + (-0.5 * (x * x))) - cos(y)))), 2.0) / (3.0 + (1.5 * (t_0 + (cos(y) * t_2))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -9.6e-5) || !(x <= 5.9e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_1 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_2 / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(t_1 * Float64(Float64(1.0 + Float64(-0.5 * Float64(x * x))) - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(cos(y) * t_2)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -9.6e-5], N[Not[LessEqual[x, 5.9e-5]], $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] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6000000000000002e-5 or 5.8999999999999998e-5 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 65.1%

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

   if -9.6000000000000002e-5 < x < 5.8999999999999998e-5

   1. Initial program 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in x around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-lft-out 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.5%

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

      1. unpow2 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-5} \lor \neg \left(x \leq 5.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 10: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := \sin y - \frac{\sin x}{16}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-6} \lor \neg \left(x \leq 5.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(t_1 \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (- (sin y) (/ (sin x) 16.0))))
   (if (or (<= x -4.8e-6) (not (<= x 5.9e-5)))
     (/
      (+ 2.0 (* (* t_1 (- (cos x) (cos y))) (* (sqrt 2.0) (sin x))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))
     (/
      (fma
       (sqrt 2.0)
       (* (+ (sin x) (* -0.0625 (sin y))) (* t_1 (- 1.0 (cos y))))
       2.0)
      (+
       3.0
       (* 1.5 (+ (+ (sqrt 5.0) -1.0) (* (cos y) (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double tmp;
	if ((x <= -4.8e-6) || !(x <= 5.9e-5)) {
		tmp = (2.0 + ((t_1 * (cos(x) - cos(y))) * (sqrt(2.0) * sin(x)))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * (t_1 * (1.0 - cos(y)))), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + -1.0) + (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	tmp = 0.0
	if ((x <= -4.8e-6) || !(x <= 5.9e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(t_1 * Float64(cos(x) - cos(y))) * Float64(sqrt(2.0) * sin(x)))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(t_1 * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + -1.0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4.8e-6], N[Not[LessEqual[x, 5.9e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(t$95$1 * N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7999999999999998e-6 or 5.8999999999999998e-5 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around 0 65.1%

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

   if -4.7999999999999998e-6 < x < 5.8999999999999998e-5

   1. Initial program 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in x around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-lft-out 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-6} \lor \neg \left(x \leq 5.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\sqrt{2} \cdot \sin x\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 11: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{5} + -1\\ t_1 := \sin y - \frac{\sin x}{16}\\ t_2 := 3 - \sqrt{5}\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-5} \lor \neg \left(x \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(t_1 \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{t_0}{2}\right) + \cos y \cdot \frac{t_2}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(t_1 \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(t_0 + \cos y \cdot t_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (sqrt 5.0) -1.0))
        (t_1 (- (sin y) (/ (sin x) 16.0)))
        (t_2 (- 3.0 (sqrt 5.0))))
   (if (or (<= x -1.85e-5) (not (<= x 6.6e-6)))
     (/
      (+ 2.0 (* (- (cos x) (cos y)) (* t_1 (* (sqrt 2.0) (sin x)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (/ t_0 2.0))) (* (cos y) (/ t_2 2.0)))))
     (/
      (fma
       (sqrt 2.0)
       (* (+ (sin x) (* -0.0625 (sin y))) (* t_1 (- 1.0 (cos y))))
       2.0)
      (+ 3.0 (* 1.5 (+ t_0 (* (cos y) t_2))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) + -1.0;
	double t_1 = sin(y) - (sin(x) / 16.0);
	double t_2 = 3.0 - sqrt(5.0);
	double tmp;
	if ((x <= -1.85e-5) || !(x <= 6.6e-6)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (t_1 * (sqrt(2.0) * sin(x))))) / (3.0 * ((1.0 + (cos(x) * (t_0 / 2.0))) + (cos(y) * (t_2 / 2.0))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * (t_1 * (1.0 - cos(y)))), 2.0) / (3.0 + (1.5 * (t_0 + (cos(y) * t_2))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) + -1.0)
	t_1 = Float64(sin(y) - Float64(sin(x) / 16.0))
	t_2 = Float64(3.0 - sqrt(5.0))
	tmp = 0.0
	if ((x <= -1.85e-5) || !(x <= 6.6e-6))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(t_1 * Float64(sqrt(2.0) * sin(x))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 / 2.0))) + Float64(cos(y) * Float64(t_2 / 2.0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(t_1 * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(t_0 + Float64(cos(y) * t_2)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.85e-5], N[Not[LessEqual[x, 6.6e-6]], $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] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(t$95$0 + N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.84999999999999991e-5 or 6.60000000000000034e-6 < x

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 65.1%

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

   if -1.84999999999999991e-5 < x < 6.60000000000000034e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in x around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-lft-out 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 99.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-5} \lor \neg \left(x \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sqrt{2} \cdot \sin x\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \frac{\sqrt{5} + -1}{2}\right) + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 12: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ \mathbf{if}\;x \leq -2600 \lor \neg \left(x \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(t_0 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)))
   (if (or (<= x -2600.0) (not (<= x 2.3e-5)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
      (* 3.0 (+ (+ 1.0 (* (cos x) (- t_0 0.5))) (* (cos y) (- 1.5 t_0)))))
     (/
      (fma
       (sqrt 2.0)
       (*
        (+ (sin x) (* -0.0625 (sin y)))
        (* (- (sin y) (/ (sin x) 16.0)) (- 1.0 (cos y))))
       2.0)
      (+
       3.0
       (* 1.5 (+ (+ (sqrt 5.0) -1.0) (* (cos y) (- 3.0 (sqrt 5.0))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double tmp;
	if ((x <= -2600.0) || !(x <= 2.3e-5)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * ((1.0 + (cos(x) * (t_0 - 0.5))) + (cos(y) * (1.5 - t_0))));
	} else {
		tmp = fma(sqrt(2.0), ((sin(x) + (-0.0625 * sin(y))) * ((sin(y) - (sin(x) / 16.0)) * (1.0 - cos(y)))), 2.0) / (3.0 + (1.5 * ((sqrt(5.0) + -1.0) + (cos(y) * (3.0 - sqrt(5.0))))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	tmp = 0.0
	if ((x <= -2600.0) || !(x <= 2.3e-5))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_0)))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(Float64(sin(x) + Float64(-0.0625 * sin(y))) * Float64(Float64(sin(y) - Float64(sin(x) / 16.0)) * Float64(1.0 - cos(y)))), 2.0) / Float64(3.0 + Float64(1.5 * Float64(Float64(sqrt(5.0) + -1.0) + Float64(cos(y) * Float64(3.0 - sqrt(5.0)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, If[Or[LessEqual[x, -2600.0], N[Not[LessEqual[x, 2.3e-5]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] + N[(-0.0625 * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(1.5 * N[(N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(3.0 - N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2600 or 2.3e-5 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 99.0%

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

   3. Simplified 99.0%

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

      1. add-exp-log_binary64 75.5%

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

   5. Applied rewrite-once 75.5%

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

      1. rem-exp-log 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0 61.9%

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

   if -2600 < x < 2.3e-5

   1. Initial program 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

   4. Taylor expanded in x around 0 98.9%

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

      1. sub-neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. distribute-lft-out 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in x around 0 99.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2600 \lor \neg \left(x \leq 2.3 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(1 - \cos y\right)\right), 2\right)}{3 + 1.5 \cdot \left(\left(\sqrt{5} + -1\right) + \cos y \cdot \left(3 - \sqrt{5}\right)\right)}\\ \end{array} \]

Alternative 13: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.205 \lor \neg \left(y \leq 0.0275\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.205) (not (<= y 0.0275)))
   (/
    (+
     2.0
     (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin y) 2.0)))))
    (*
     3.0
     (+
      (+ 1.0 (* (cos x) (- (/ (sqrt 5.0) 2.0) 0.5)))
      (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5)))))))
   (/
    (fma (sqrt 2.0) (* -0.0625 (* (pow (sin x) 2.0) (+ (cos x) -1.0))) 2.0)
    (+
     3.0
     (fma
      (cos y)
      (/ (/ 4.0 (+ 3.0 (sqrt 5.0))) 0.6666666666666666)
      (/ (* (cos x) (+ (sqrt 5.0) -1.0)) 0.6666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.205) || !(y <= 0.0275)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * ((1.0 + (cos(x) * ((sqrt(5.0) / 2.0) - 0.5))) + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5))))));
	} else {
		tmp = fma(sqrt(2.0), (-0.0625 * (pow(sin(x), 2.0) * (cos(x) + -1.0))), 2.0) / (3.0 + fma(cos(y), ((4.0 / (3.0 + sqrt(5.0))) / 0.6666666666666666), ((cos(x) * (sqrt(5.0) + -1.0)) / 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.205) || !(y <= 0.0275))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(y) ^ 2.0))))) / Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(Float64(sqrt(5.0) / 2.0) - 0.5))) + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5)))))));
	else
		tmp = Float64(fma(sqrt(2.0), Float64(-0.0625 * Float64((sin(x) ^ 2.0) * Float64(cos(x) + -1.0))), 2.0) / Float64(3.0 + fma(cos(y), Float64(Float64(4.0 / Float64(3.0 + sqrt(5.0))) / 0.6666666666666666), Float64(Float64(cos(x) * Float64(sqrt(5.0) + -1.0)) / 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.205], N[Not[LessEqual[y, 0.0275]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[(3.0 + N[(N[Cos[y], $MachinePrecision] * N[(N[(4.0 / N[(3.0 + N[Sqrt[5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision] + N[(N[(N[Cos[x], $MachinePrecision] * N[(N[Sqrt[5.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.204999999999999988 or 0.0275000000000000001 < y

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 99.0%

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

   3. Simplified 99.0%

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

      1. add-exp-log_binary64 24.1%

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

   5. Applied rewrite-once 24.1%

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

      1. rem-exp-log 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0 59.8%

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

      1. flip-- 59.6%

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

      2. sub-neg 59.6%

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

      3. metadata-eval 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{\color{blue}{2.25} + \left(-\frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      4. div-inv 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      5. metadata-eval 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      6. div-inv 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      7. metadata-eval 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      8. swap-sqr 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      9. rem-square-sqrt 59.7%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      10. metadata-eval 59.7%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-5 \cdot \color{blue}{0.25}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      11. metadata-eval 59.7%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{1.25}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      12. metadata-eval 59.7%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      13. metadata-eval 59.7%

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

      14. div-inv 59.7%

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

      15. metadata-eval 59.7%

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

   10. Applied egg-rr 59.8%

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

   if -0.204999999999999988 < y < 0.0275000000000000001

   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 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]
   4. Step-by-step derivation

      1. flip-- 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      2. sub-neg 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{3 \cdot 3 + \left(-\sqrt{5} \cdot \sqrt{5}\right)}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{9} + \left(-\sqrt{5} \cdot \sqrt{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      4. rem-square-sqrt 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \left(-\color{blue}{5}\right)}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      5. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{9 + \color{blue}{-5}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{\color{blue}{4}}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

      7. +-commutative 99.7%

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{\color{blue}{\sqrt{5} + 3}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\color{blue}{\frac{4}{\sqrt{5} + 3}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)} \]

   6. Taylor expanded in y around 0 97.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.205 \lor \neg \left(y \leq 0.0275\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{2}, -0.0625 \cdot \left({\sin x}^{2} \cdot \left(\cos x + -1\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{\frac{4}{3 + \sqrt{5}}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 14: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 1 + \cos x \cdot \left(t_0 - 0.5\right)\\ \mathbf{if}\;x \leq -86000000000000 \lor \neg \left(x \leq 3350000\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (+ 1.0 (* (cos x) (- t_0 0.5)))))
   (if (or (<= x -86000000000000.0) (not (<= x 3350000.0)))
     (/
      (+
       2.0
       (* (- (cos x) (cos y)) (* -0.0625 (* (sqrt 2.0) (pow (sin x) 2.0)))))
      (* 3.0 (+ t_1 (* (cos y) (- 1.5 t_0)))))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (* 3.0 (+ t_1 (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = 1.0 + (cos(x) * (t_0 - 0.5));
	double tmp;
	if ((x <= -86000000000000.0) || !(x <= 3350000.0)) {
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(x), 2.0))))) / (3.0 * (t_1 + (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 * (t_1 + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = 1.0d0 + (cos(x) * (t_0 - 0.5d0))
    if ((x <= (-86000000000000.0d0)) .or. (.not. (x <= 3350000.0d0))) then
        tmp = (2.0d0 + ((cos(x) - cos(y)) * ((-0.0625d0) * (sqrt(2.0d0) * (sin(x) ** 2.0d0))))) / (3.0d0 * (t_1 + (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 * (t_1 + (cos(y) * (1.0d0 / (1.5d0 + (sqrt(5.0d0) * 0.5d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = 1.0 + (Math.cos(x) * (t_0 - 0.5));
	double tmp;
	if ((x <= -86000000000000.0) || !(x <= 3350000.0)) {
		tmp = (2.0 + ((Math.cos(x) - Math.cos(y)) * (-0.0625 * (Math.sqrt(2.0) * Math.pow(Math.sin(x), 2.0))))) / (3.0 * (t_1 + (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 * (t_1 + (Math.cos(y) * (1.0 / (1.5 + (Math.sqrt(5.0) * 0.5))))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = 1.0 + (math.cos(x) * (t_0 - 0.5))
	tmp = 0
	if (x <= -86000000000000.0) or not (x <= 3350000.0):
		tmp = (2.0 + ((math.cos(x) - math.cos(y)) * (-0.0625 * (math.sqrt(2.0) * math.pow(math.sin(x), 2.0))))) / (3.0 * (t_1 + (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 * (t_1 + (math.cos(y) * (1.0 / (1.5 + (math.sqrt(5.0) * 0.5))))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5)))
	tmp = 0.0
	if ((x <= -86000000000000.0) || !(x <= 3350000.0))
		tmp = Float64(Float64(2.0 + Float64(Float64(cos(x) - cos(y)) * Float64(-0.0625 * Float64(sqrt(2.0) * (sin(x) ^ 2.0))))) / Float64(3.0 * Float64(t_1 + 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(t_1 + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = 1.0 + (cos(x) * (t_0 - 0.5));
	tmp = 0.0;
	if ((x <= -86000000000000.0) || ~((x <= 3350000.0)))
		tmp = (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * (sin(x) ^ 2.0))))) / (3.0 * (t_1 + (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 * (t_1 + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -86000000000000.0], N[Not[LessEqual[x, 3350000.0]], $MachinePrecision]], N[(N[(2.0 + N[(N[(N[Cos[x], $MachinePrecision] - N[Cos[y], $MachinePrecision]), $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $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[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.6e13 or 3.35e6 < x

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 99.0%

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

   3. Simplified 99.0%

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

      1. add-exp-log_binary64 76.5%

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

   5. Applied rewrite-once 76.5%

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

      1. rem-exp-log 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0 62.1%

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

   if -8.6e13 < x < 3.35e6

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-rgt-in 99.7%

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

      3. cos-neg 99.7%

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

      4. distribute-rgt-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. flip-- 97.4%

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

      2. sub-neg 97.4%

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

      3. metadata-eval 97.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{\color{blue}{2.25} + \left(-\frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      4. div-inv 97.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      5. metadata-eval 97.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      6. div-inv 97.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      7. metadata-eval 97.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      8. swap-sqr 97.4%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      9. rem-square-sqrt 97.5%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      10. metadata-eval 97.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-5 \cdot \color{blue}{0.25}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      11. metadata-eval 97.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{1.25}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      12. metadata-eval 97.5%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      13. metadata-eval 97.5%

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

      14. div-inv 97.5%

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

      15. metadata-eval 97.5%

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

   8. Applied egg-rr 97.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -86000000000000 \lor \neg \left(x \leq 3350000\right):\\ \;\;\;\;\frac{2 + \left(\cos x - \cos y\right) \cdot \left(-0.0625 \cdot \left(\sqrt{2} \cdot {\sin x}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \end{array} \]

Alternative 15: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{5}}{2}\\ t_1 := 1 + \cos x \cdot \left(t_0 - 0.5\right)\\ \mathbf{if}\;y \leq -0.205 \lor \neg \left(y \leq 0.0275\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(t_1 + \cos y \cdot \left(1.5 - t_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt 5.0) 2.0)) (t_1 (+ 1.0 (* (cos x) (- t_0 0.5)))))
   (if (or (<= y -0.205) (not (<= y 0.0275)))
     (/
      (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
      (* 3.0 (+ t_1 (* (cos y) (/ 1.0 (+ 1.5 (* (sqrt 5.0) 0.5)))))))
     (/
      (+ 2.0 (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
      (* 3.0 (+ t_1 (* (cos y) (- 1.5 t_0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(5.0) / 2.0;
	double t_1 = 1.0 + (cos(x) * (t_0 - 0.5));
	double tmp;
	if ((y <= -0.205) || !(y <= 0.0275)) {
		tmp = (2.0 + (-0.0625 * (pow(sin(y), 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (t_1 + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5))))));
	} else {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 * (t_1 + (cos(y) * (1.5 - t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(5.0d0) / 2.0d0
    t_1 = 1.0d0 + (cos(x) * (t_0 - 0.5d0))
    if ((y <= (-0.205d0)) .or. (.not. (y <= 0.0275d0))) then
        tmp = (2.0d0 + ((-0.0625d0) * ((sin(y) ** 2.0d0) * (sqrt(2.0d0) * (1.0d0 - cos(y)))))) / (3.0d0 * (t_1 + (cos(y) * (1.0d0 / (1.5d0 + (sqrt(5.0d0) * 0.5d0))))))
    else
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / (3.0d0 * (t_1 + (cos(y) * (1.5d0 - t_0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(5.0) / 2.0;
	double t_1 = 1.0 + (Math.cos(x) * (t_0 - 0.5));
	double tmp;
	if ((y <= -0.205) || !(y <= 0.0275)) {
		tmp = (2.0 + (-0.0625 * (Math.pow(Math.sin(y), 2.0) * (Math.sqrt(2.0) * (1.0 - Math.cos(y)))))) / (3.0 * (t_1 + (Math.cos(y) * (1.0 / (1.5 + (Math.sqrt(5.0) * 0.5))))));
	} else {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / (3.0 * (t_1 + (Math.cos(y) * (1.5 - t_0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(5.0) / 2.0
	t_1 = 1.0 + (math.cos(x) * (t_0 - 0.5))
	tmp = 0
	if (y <= -0.205) or not (y <= 0.0275):
		tmp = (2.0 + (-0.0625 * (math.pow(math.sin(y), 2.0) * (math.sqrt(2.0) * (1.0 - math.cos(y)))))) / (3.0 * (t_1 + (math.cos(y) * (1.0 / (1.5 + (math.sqrt(5.0) * 0.5))))))
	else:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / (3.0 * (t_1 + (math.cos(y) * (1.5 - t_0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(5.0) / 2.0)
	t_1 = Float64(1.0 + Float64(cos(x) * Float64(t_0 - 0.5)))
	tmp = 0.0
	if ((y <= -0.205) || !(y <= 0.0275))
		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(t_1 + Float64(cos(y) * Float64(1.0 / Float64(1.5 + Float64(sqrt(5.0) * 0.5)))))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / Float64(3.0 * Float64(t_1 + Float64(cos(y) * Float64(1.5 - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(5.0) / 2.0;
	t_1 = 1.0 + (cos(x) * (t_0 - 0.5));
	tmp = 0.0;
	if ((y <= -0.205) || ~((y <= 0.0275)))
		tmp = (2.0 + (-0.0625 * ((sin(y) ^ 2.0) * (sqrt(2.0) * (1.0 - cos(y)))))) / (3.0 * (t_1 + (cos(y) * (1.0 / (1.5 + (sqrt(5.0) * 0.5))))));
	else
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / (3.0 * (t_1 + (cos(y) * (1.5 - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -0.205], N[Not[LessEqual[y, 0.0275]], $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[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.0 / N[(1.5 + N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(t$95$1 + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.204999999999999988 or 0.0275000000000000001 < y

   1. Initial program 99.0%

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

      1. associate-*l* 99.0%

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

      2. distribute-rgt-in 98.9%

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

      3. cos-neg 98.9%

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

      4. distribute-rgt-in 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in x around 0 59.6%

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

      1. *-commutative 59.6%

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

   6. Simplified 59.6%

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

      1. flip-- 59.6%

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

      2. sub-neg 59.6%

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

      3. metadata-eval 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{\color{blue}{2.25} + \left(-\frac{\sqrt{5}}{2} \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      4. div-inv 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)} \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      5. metadata-eval 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot \color{blue}{0.5}\right) \cdot \frac{\sqrt{5}}{2}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      6. div-inv 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{5} \cdot \frac{1}{2}\right)}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      7. metadata-eval 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\left(\sqrt{5} \cdot 0.5\right) \cdot \left(\sqrt{5} \cdot \color{blue}{0.5}\right)\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      8. swap-sqr 59.6%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right) \cdot \left(0.5 \cdot 0.5\right)}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      9. rem-square-sqrt 59.7%

         \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{5} \cdot \left(0.5 \cdot 0.5\right)\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      10. metadata-eval 59.7%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-5 \cdot \color{blue}{0.25}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      11. metadata-eval 59.7%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \left(-\color{blue}{1.25}\right)}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      12. metadata-eval 59.7%

          \[\leadsto \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{2.25 + \color{blue}{-1.25}}{1.5 + \frac{\sqrt{5}}{2}}\right)} \]

      13. metadata-eval 59.7%

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

      14. div-inv 59.7%

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

      15. metadata-eval 59.7%

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

   8. Applied egg-rr 59.7%

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

   if -0.204999999999999988 < y < 0.0275000000000000001

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-rgt-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-rgt-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. associate-*r* 97.3%

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

      2. *-commutative 97.3%

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

      3. sub-neg 97.3%

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

      4. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.205 \lor \neg \left(y \leq 0.0275\right):\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \frac{1}{1.5 + \sqrt{5} \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]

Alternative 16: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin y}^{2}\\ t_1 := \frac{\sqrt{5}}{2}\\ t_2 := 3 \cdot \left(\left(1 + \cos x \cdot \left(t_1 - 0.5\right)\right) + \cos y \cdot \left(1.5 - t_1\right)\right)\\ t_3 := 1 - \cos y\\ \mathbf{if}\;y \leq -0.205:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_3 \cdot \left(\sqrt{2} \cdot t_0\right)\right)}{t_2}\\ \mathbf{elif}\;y \leq 0.0275:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(t_0 \cdot \left(\sqrt{2} \cdot t_3\right)\right)}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (sin y) 2.0))
        (t_1 (/ (sqrt 5.0) 2.0))
        (t_2
         (* 3.0 (+ (+ 1.0 (* (cos x) (- t_1 0.5))) (* (cos y) (- 1.5 t_1)))))
        (t_3 (- 1.0 (cos y))))
   (if (<= y -0.205)
     (/ (+ 2.0 (* -0.0625 (* t_3 (* (sqrt 2.0) t_0)))) t_2)
     (if (<= y 0.0275)
       (/
        (+
         2.0
         (* (* -0.0625 (pow (sin x) 2.0)) (* (sqrt 2.0) (+ (cos x) -1.0))))
        t_2)
       (/ (+ 2.0 (* -0.0625 (* t_0 (* (sqrt 2.0) t_3)))) t_2)))))

C

double code(double x, double y) {
	double t_0 = pow(sin(y), 2.0);
	double t_1 = sqrt(5.0) / 2.0;
	double t_2 = 3.0 * ((1.0 + (cos(x) * (t_1 - 0.5))) + (cos(y) * (1.5 - t_1)));
	double t_3 = 1.0 - cos(y);
	double tmp;
	if (y <= -0.205) {
		tmp = (2.0 + (-0.0625 * (t_3 * (sqrt(2.0) * t_0)))) / t_2;
	} else if (y <= 0.0275) {
		tmp = (2.0 + ((-0.0625 * pow(sin(x), 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / t_2;
	} else {
		tmp = (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * t_3)))) / t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(y) ** 2.0d0
    t_1 = sqrt(5.0d0) / 2.0d0
    t_2 = 3.0d0 * ((1.0d0 + (cos(x) * (t_1 - 0.5d0))) + (cos(y) * (1.5d0 - t_1)))
    t_3 = 1.0d0 - cos(y)
    if (y <= (-0.205d0)) then
        tmp = (2.0d0 + ((-0.0625d0) * (t_3 * (sqrt(2.0d0) * t_0)))) / t_2
    else if (y <= 0.0275d0) then
        tmp = (2.0d0 + (((-0.0625d0) * (sin(x) ** 2.0d0)) * (sqrt(2.0d0) * (cos(x) + (-1.0d0))))) / t_2
    else
        tmp = (2.0d0 + ((-0.0625d0) * (t_0 * (sqrt(2.0d0) * t_3)))) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.sin(y), 2.0);
	double t_1 = Math.sqrt(5.0) / 2.0;
	double t_2 = 3.0 * ((1.0 + (Math.cos(x) * (t_1 - 0.5))) + (Math.cos(y) * (1.5 - t_1)));
	double t_3 = 1.0 - Math.cos(y);
	double tmp;
	if (y <= -0.205) {
		tmp = (2.0 + (-0.0625 * (t_3 * (Math.sqrt(2.0) * t_0)))) / t_2;
	} else if (y <= 0.0275) {
		tmp = (2.0 + ((-0.0625 * Math.pow(Math.sin(x), 2.0)) * (Math.sqrt(2.0) * (Math.cos(x) + -1.0)))) / t_2;
	} else {
		tmp = (2.0 + (-0.0625 * (t_0 * (Math.sqrt(2.0) * t_3)))) / t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(math.sin(y), 2.0)
	t_1 = math.sqrt(5.0) / 2.0
	t_2 = 3.0 * ((1.0 + (math.cos(x) * (t_1 - 0.5))) + (math.cos(y) * (1.5 - t_1)))
	t_3 = 1.0 - math.cos(y)
	tmp = 0
	if y <= -0.205:
		tmp = (2.0 + (-0.0625 * (t_3 * (math.sqrt(2.0) * t_0)))) / t_2
	elif y <= 0.0275:
		tmp = (2.0 + ((-0.0625 * math.pow(math.sin(x), 2.0)) * (math.sqrt(2.0) * (math.cos(x) + -1.0)))) / t_2
	else:
		tmp = (2.0 + (-0.0625 * (t_0 * (math.sqrt(2.0) * t_3)))) / t_2
	return tmp

Julia

function code(x, y)
	t_0 = sin(y) ^ 2.0
	t_1 = Float64(sqrt(5.0) / 2.0)
	t_2 = Float64(3.0 * Float64(Float64(1.0 + Float64(cos(x) * Float64(t_1 - 0.5))) + Float64(cos(y) * Float64(1.5 - t_1))))
	t_3 = Float64(1.0 - cos(y))
	tmp = 0.0
	if (y <= -0.205)
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_3 * Float64(sqrt(2.0) * t_0)))) / t_2);
	elseif (y <= 0.0275)
		tmp = Float64(Float64(2.0 + Float64(Float64(-0.0625 * (sin(x) ^ 2.0)) * Float64(sqrt(2.0) * Float64(cos(x) + -1.0)))) / t_2);
	else
		tmp = Float64(Float64(2.0 + Float64(-0.0625 * Float64(t_0 * Float64(sqrt(2.0) * t_3)))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) ^ 2.0;
	t_1 = sqrt(5.0) / 2.0;
	t_2 = 3.0 * ((1.0 + (cos(x) * (t_1 - 0.5))) + (cos(y) * (1.5 - t_1)));
	t_3 = 1.0 - cos(y);
	tmp = 0.0;
	if (y <= -0.205)
		tmp = (2.0 + (-0.0625 * (t_3 * (sqrt(2.0) * t_0)))) / t_2;
	elseif (y <= 0.0275)
		tmp = (2.0 + ((-0.0625 * (sin(x) ^ 2.0)) * (sqrt(2.0) * (cos(x) + -1.0)))) / t_2;
	else
		tmp = (2.0 + (-0.0625 * (t_0 * (sqrt(2.0) * t_3)))) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(3.0 * N[(N[(1.0 + N[(N[Cos[x], $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[(1.5 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Cos[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.205], N[(N[(2.0 + N[(-0.0625 * N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 0.0275], N[(N[(2.0 + N[(N[(-0.0625 * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 + N[(-0.0625 * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.204999999999999988

   1. Initial program 99.2%

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

      1. associate-*l* 99.2%

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

      2. distribute-rgt-in 99.0%

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

      3. cos-neg 99.0%

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

      4. distribute-rgt-in 99.2%

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

   3. Simplified 99.2%

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

      1. add-exp-log_binary64 26.5%

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

   5. Applied rewrite-once 26.5%

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

      1. rem-exp-log 99.2%

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

      2. associate-*r* 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*l* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 54.8%

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

      1. associate-*r* 54.8%

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

   10. Simplified 54.8%

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

   if -0.204999999999999988 < y < 0.0275000000000000001

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. distribute-rgt-in 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-rgt-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. associate-*r* 97.3%

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

      2. *-commutative 97.3%

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

      3. sub-neg 97.3%

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

      4. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

   if 0.0275000000000000001 < y

   1. Initial program 98.8%

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

      1. associate-*l* 98.8%

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

      2. distribute-rgt-in 98.8%

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

      3. cos-neg 98.8%

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

      4. distribute-rgt-in 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 64.2%

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

      1. *-commutative 64.2%

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

   6. Simplified 64.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.205:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left(\left(1 - \cos y\right) \cdot \left(\sqrt{2} \cdot {\sin y}^{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{elif}\;y \leq 0.0275:\\ \;\;\;\;\frac{2 + \left(-0.0625 \cdot {\sin x}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\cos x + -1\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}\\ \end{array} \]

Alternative 17: 63.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

   1. add-exp-log_binary64 65.7%

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

5. Applied rewrite-once 65.7%

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

   1. rem-exp-log 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around 0 63.2%

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

   1. associate-*r* 63.2%

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

10. Simplified 63.2%

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

11. Final simplification 63.2%

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

Alternative 18: 63.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 63.2%

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

   1. *-commutative 63.2%

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

6. Simplified 63.2%

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

7. Final simplification 63.2%

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

Alternative 19: 60.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (2.0 + ((cos(x) - cos(y)) * (-0.0625 * (sqrt(2.0) * pow(sin(y), 2.0))))) / (3.0 * ((cos(y) * (1.5 - (sqrt(5.0) / 2.0))) + (1.0 + ((sqrt(5.0) * 0.5) - 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)) * ((-0.0625d0) * (sqrt(2.0d0) * (sin(y) ** 2.0d0))))) / (3.0d0 * ((cos(y) * (1.5d0 - (sqrt(5.0d0) / 2.0d0))) + (1.0d0 + ((sqrt(5.0d0) * 0.5d0) - 0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

   1. add-exp-log_binary64 65.7%

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

5. Applied rewrite-once 65.7%

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

   1. rem-exp-log 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around 0 63.3%

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

9. Taylor expanded in x around 0 60.2%

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

10. Final simplification 60.2%

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

Alternative 20: 60.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{2}, \left(\sin x + -0.0625 \cdot \sin y\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right), 2\right)}{3 + \mathsf{fma}\left(\cos y, \frac{3 - \sqrt{5}}{0.6666666666666666}, \frac{\cos x \cdot \left(\sqrt{5} + -1\right)}{0.6666666666666666}\right)}} \]

4. Taylor expanded in x around 0 60.7%

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

   1. sub-neg 60.7%

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

   2. metadata-eval 60.7%

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

   3. distribute-lft-out 60.7%

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

6. Simplified 60.7%

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

7. Taylor expanded in x around 0 60.2%

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

   1. associate-*r* 60.2%

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

9. Simplified 60.2%

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

10. Final simplification 60.2%

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

Alternative 21: 60.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := 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[(N[Cos[y], $MachinePrecision] * N[(1.5 - N[(N[Sqrt[5.0], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(N[Sqrt[5.0], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 63.2%

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

   1. *-commutative 63.2%

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

6. Simplified 63.2%

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

7. Taylor expanded in x around 0 60.2%

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

8. Final simplification 60.2%

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

Alternative 22: 16.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \langle \left( \begin{array}{l} t_0 := 0.5 \cdot \sqrt{5}\\ 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\cos x \cdot \left(t_0 - 0.5\right) + \cos y \cdot \left(1.5 - t_0\right)\right)} \end{array} \right)_{\text{binary32}} \rangle_{\text{binary64}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (cast
  (!
   :precision
   binary32
   (let* ((t_0 (* 0.5 (sqrt 5.0))))
     (*
      0.3333333333333333
      (/
       (+ 2.0 (* -0.0625 (* (pow (sin y) 2.0) (* (sqrt 2.0) (- 1.0 (cos y))))))
       (+ 1.0 (+ (* (cos x) (- t_0 0.5)) (* (cos y) (- 1.5 t_0))))))))))

C

double code(double x, double y) {
	float t_0_1 = 0.5f * sqrtf(5.0f);
	float tmp = 0.3333333333333333f * ((2.0f + (-0.0625f * (powf(sinf(y), 2.0f) * (sqrtf(2.0f) * (1.0f - cosf(y)))))) / (1.0f + ((cosf(x) * (t_0_1 - 0.5f)) + (cosf(y) * (1.5f - t_0_1)))));
	return (double) tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(4) :: t_0
    real(4) :: t_0_1
    real(4) :: tmp
    t_0_1 = 0.5e0 * sqrt(5.0e0)
    tmp = 0.3333333333333333e0 * ((2.0e0 + ((-0.0625e0) * ((sin(y) ** 2.0e0) * (sqrt(2.0e0) * (1.0e0 - cos(y)))))) / (1.0e0 + ((cos(x) * (t_0_1 - 0.5e0)) + (cos(y) * (1.5e0 - t_0_1)))))
    code = real(tmp, 8)
end function

Julia

function code(x, y)
	t_0_1 = Float32(Float32(0.5) * sqrt(Float32(5.0)))
	tmp = Float32(Float32(0.3333333333333333) * Float32(Float32(Float32(2.0) + Float32(Float32(-0.0625) * Float32((sin(y) ^ Float32(2.0)) * Float32(sqrt(Float32(2.0)) * Float32(Float32(1.0) - cos(y)))))) / Float32(Float32(1.0) + Float32(Float32(cos(x) * Float32(t_0_1 - Float32(0.5))) + Float32(cos(y) * Float32(Float32(1.5) - t_0_1))))))
	return Float64(tmp)
end

MATLAB

function tmp_3 = code(x, y)
	t_0_2 = single(0.5) * sqrt(single(5.0));
	tmp = single(0.3333333333333333) * ((single(2.0) + (single(-0.0625) * ((sin(y) ^ single(2.0)) * (sqrt(single(2.0)) * (single(1.0) - cos(y)))))) / (single(1.0) + ((cos(x) * (t_0_2 - single(0.5))) + (cos(y) * (single(1.5) - t_0_2)))));
	tmp_3 = double(tmp);
end

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. cos-neg 99.3%

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

   4. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in x around 0 63.2%

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

   1. *-commutative 63.2%

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

6. Simplified 63.2%

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

   1. rewrite-binary64/binary32 18.5%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

8. Applied rewrite-once 18.5%

   \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\left(1 - \cos y\right) \cdot \sqrt{2}\right)\right)}{3 \cdot \left(\left(1 + \cos x \cdot \left(\frac{\sqrt{5}}{2} - 0.5\right)\right) + \cos y \cdot \left(1.5 - \frac{\sqrt{5}}{2}\right)\right)}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

9. Taylor expanded in y around 0 18.5%

   \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right)}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

10. Final simplification 18.5%

    \[\leadsto \langle \left( 0.3333333333333333 \cdot \frac{2 + -0.0625 \cdot \left({\sin y}^{2} \cdot \left(\sqrt{2} \cdot \left(1 - \cos y\right)\right)\right)}{1 + \left(\cos x \cdot \left(0.5 \cdot \sqrt{5} - 0.5\right) + \cos y \cdot \left(1.5 - 0.5 \cdot \sqrt{5}\right)\right)} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

Reproduce

herbie shell --seed 2023297 
(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))))))